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