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Lemmas fiberwise_{finite,countable}_preimage #1382

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Lemmas fiberwise_{finite,countable}_preimage #1382

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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -4,6 +4,9 @@

### Added

- in file `cardinality.v`,
+ lemmas `fiberwise_finite_preimage`, `fiberwise_countable_preimage`

- file `Rstruct_topology.v`

- package `coq-mathcomp-reals` depending on `coq-mathcomp-classical`
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18 changes: 18 additions & 0 deletions classical/cardinality.v
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Expand Up @@ -840,6 +840,15 @@ have Gy : y \in G k by rewrite in_fset_set ?inE//; apply: Ffin.
by exists (Tagged G [` Gy]%fset).
Qed.

Lemma fiberwise_finite_preimage {T U} (B : set U) (f : T -> U) :
finite_set B -> (forall b, B b -> finite_set (f @^-1` [set b])) ->
finite_set (f @^-1` B).
Proof.
move=> *.
rewrite -(image_id B) -bigcup_imset1 preimage_bigcup.
exact: bigcup_finite.
Qed.

Lemma trivIset_sum_card (T : choiceType) (F : nat -> set T) n :
(forall n, finite_set (F n)) -> trivIset [set: nat] F ->
(\sum_(i < n) #|` fset_set (F i)| =
Expand Down Expand Up @@ -1143,6 +1152,15 @@ apply/pcard_geP/surjPex; exists (fun (k : {i & G i}) => val (projT2 k)).
by move=> x [i _] Gix/=; exists (Tagged G (SigSub (mem_set Gix))).
Qed.

Lemma fiberwise_countable_preimage {T U} (B : set U) (f : T -> U) :
countable B -> (forall b, B b -> countable (f @^-1` [set b])) ->
countable (f @^-1` B).
Proof.
move=> *.
rewrite -(image_id B) -bigcup_imset1 preimage_bigcup.
exact: bigcup_countable.
Qed.

Lemma countableXR T T' (A : set T) (B : T -> set T') :
countable A -> (forall i, A i -> countable (B i)) -> countable (A `*`` B).
Proof.
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