This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem funcestrcsetclem5

Description: Lemma 5 for funcestrcsetc . (Contributed by AV, 23-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
funcestrcsetc.m ⊢ M = Base X
funcestrcsetc.n ⊢ N = Base Y
Assertion funcestrcsetclem5 ⊢ φ ∧ X ∈ B ∧ Y ∈ B → X G Y = I ↾ N M

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 funcestrcsetc.m ⊢ M = Base X
9 funcestrcsetc.n ⊢ N = Base Y
10 7 adantr ⊢ φ ∧ X ∈ B ∧ Y ∈ B → G = x ∈ B , y ∈ B ⟼ I ↾ Base y Base x
11 fveq2 ⊢ y = Y → Base y = Base Y
12 fveq2 ⊢ x = X → Base x = Base X
13 11 12 oveqan12rd ⊢ x = X ∧ y = Y → Base y Base x = Base Y Base X
14 9 8 oveq12i ⊢ N M = Base Y Base X
15 13 14 eqtr4di ⊢ x = X ∧ y = Y → Base y Base x = N M
16 15 reseq2d ⊢ x = X ∧ y = Y → I ↾ Base y Base x = I ↾ N M
17 16 adantl ⊢ φ ∧ X ∈ B ∧ Y ∈ B ∧ x = X ∧ y = Y → I ↾ Base y Base x = I ↾ N M
18 simprl ⊢ φ ∧ X ∈ B ∧ Y ∈ B → X ∈ B
19 simprr ⊢ φ ∧ X ∈ B ∧ Y ∈ B → Y ∈ B
20 ovexd ⊢ φ ∧ X ∈ B ∧ Y ∈ B → N M ∈ V
21 20 resiexd ⊢ φ ∧ X ∈ B ∧ Y ∈ B → I ↾ N M ∈ V
22 10 17 18 19 21 ovmpod ⊢ φ ∧ X ∈ B ∧ Y ∈ B → X G Y = I ↾ N M