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Metamath Proof Explorer

Theorem funcestrcsetclem4

Description: Lemma 4 for funcestrcsetc . (Contributed by AV, 22-Mar-2020)

Ref Expression
Hypotheses funcestrcsetc.e ⊢ E = ExtStrCat ⁡ U
funcestrcsetc.s ⊢ S = SetCat ⁡ U
funcestrcsetc.b ⊢ B = Base E
funcestrcsetc.c ⊢ C = Base S
funcestrcsetc.u ⊢ φ → U ∈ WUni
funcestrcsetc.f ⊢ φ → F = x ∈ B ⟼ Base x
funcestrcsetc.g ⊢ φ → G = x ∈ B , y ∈ B ⟼ I ↾ Base y Base x
Assertion funcestrcsetclem4 ⊢ φ → G Fn B × B

Proof

Step Hyp Ref Expression
1 funcestrcsetc.e ⊢ E = ExtStrCat ⁡ U
2 funcestrcsetc.s ⊢ S = SetCat ⁡ U
3 funcestrcsetc.b ⊢ B = Base E
4 funcestrcsetc.c ⊢ C = Base S
5 funcestrcsetc.u ⊢ φ → U ∈ WUni
6 funcestrcsetc.f ⊢ φ → F = x ∈ B ⟼ Base x
7 funcestrcsetc.g ⊢ φ → G = x ∈ B , y ∈ B ⟼ I ↾ Base y Base x
8 eqid ⊢ x ∈ B , y ∈ B ⟼ I ↾ Base y Base x = x ∈ B , y ∈ B ⟼ I ↾ Base y Base x
9 ovex ⊢ Base y Base x ∈ V
10 resiexg ⊢ Base y Base x ∈ V → I ↾ Base y Base x ∈ V
11 9 10 ax-mp ⊢ I ↾ Base y Base x ∈ V
12 8 11 fnmpoi ⊢ x ∈ B , y ∈ B ⟼ I ↾ Base y Base x Fn B × B
13 7 fneq1d ⊢ φ → G Fn B × B ↔ x ∈ B , y ∈ B ⟼ I ↾ Base y Base x Fn B × B
14 12 13 mpbiri ⊢ φ → G Fn B × B