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Metamath Proof Explorer

Theorem funcestrcsetclem7

Description: Lemma 7 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 
|- ( ph -> U e. WUni )
funcestrcsetc.f 
|- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
funcestrcsetc.g 
|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )
Assertion funcestrcsetclem7 
|- ( ( ph /\ X e. B ) -> ( ( X G X ) ` ( ( Id ` E ) ` X ) ) = ( ( Id ` S ) ` ( F ` X ) ) )

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 
 |-  ( ph -> U e. WUni )
6 funcestrcsetc.f 
 |-  ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
7 funcestrcsetc.g 
 |-  ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )
8 eqid 
 |-  ( Base ` X ) = ( Base ` X )
9 1 2 3 4 5 6 7 8 8 funcestrcsetclem5 
 |-  ( ( ph /\ ( X e. B /\ X e. B ) ) -> ( X G X ) = ( _I |` ( ( Base ` X ) ^m ( Base ` X ) ) ) )
10 9 anabsan2 
 |-  ( ( ph /\ X e. B ) -> ( X G X ) = ( _I |` ( ( Base ` X ) ^m ( Base ` X ) ) ) )
11 eqid 
 |-  ( Id ` E ) = ( Id ` E )
12 5 adantr 
 |-  ( ( ph /\ X e. B ) -> U e. WUni )
13 1 5 estrcbas 
 |-  ( ph -> U = ( Base ` E ) )
14 3 13 eqtr4id 
 |-  ( ph -> B = U )
15 14 eleq2d 
 |-  ( ph -> ( X e. B <-> X e. U ) )
16 15 biimpa 
 |-  ( ( ph /\ X e. B ) -> X e. U )
17 1 11 12 16 estrcid 
 |-  ( ( ph /\ X e. B ) -> ( ( Id ` E ) ` X ) = ( _I |` ( Base ` X ) ) )
18 10 17 fveq12d 
 |-  ( ( ph /\ X e. B ) -> ( ( X G X ) ` ( ( Id ` E ) ` X ) ) = ( ( _I |` ( ( Base ` X ) ^m ( Base ` X ) ) ) ` ( _I |` ( Base ` X ) ) ) )
19 fvex 
 |-  ( Base ` X ) e. _V
20 19 19 pm3.2i 
 |-  ( ( Base ` X ) e. _V /\ ( Base ` X ) e. _V )
21 20 a1i 
 |-  ( ( ph /\ X e. B ) -> ( ( Base ` X ) e. _V /\ ( Base ` X ) e. _V ) )
22 f1oi 
 |-  ( _I |` ( Base ` X ) ) : ( Base ` X ) -1-1-onto-> ( Base ` X )
23 f1of 
 |-  ( ( _I |` ( Base ` X ) ) : ( Base ` X ) -1-1-onto-> ( Base ` X ) -> ( _I |` ( Base ` X ) ) : ( Base ` X ) --> ( Base ` X ) )
24 22 23 ax-mp 
 |-  ( _I |` ( Base ` X ) ) : ( Base ` X ) --> ( Base ` X )
25 elmapg 
 |-  ( ( ( Base ` X ) e. _V /\ ( Base ` X ) e. _V ) -> ( ( _I |` ( Base ` X ) ) e. ( ( Base ` X ) ^m ( Base ` X ) ) <-> ( _I |` ( Base ` X ) ) : ( Base ` X ) --> ( Base ` X ) ) )
26 24 25 mpbiri 
 |-  ( ( ( Base ` X ) e. _V /\ ( Base ` X ) e. _V ) -> ( _I |` ( Base ` X ) ) e. ( ( Base ` X ) ^m ( Base ` X ) ) )
27 fvresi 
 |-  ( ( _I |` ( Base ` X ) ) e. ( ( Base ` X ) ^m ( Base ` X ) ) -> ( ( _I |` ( ( Base ` X ) ^m ( Base ` X ) ) ) ` ( _I |` ( Base ` X ) ) ) = ( _I |` ( Base ` X ) ) )
28 21 26 27 3syl 
 |-  ( ( ph /\ X e. B ) -> ( ( _I |` ( ( Base ` X ) ^m ( Base ` X ) ) ) ` ( _I |` ( Base ` X ) ) ) = ( _I |` ( Base ` X ) ) )
29 1 2 3 4 5 6 funcestrcsetclem1 
 |-  ( ( ph /\ X e. B ) -> ( F ` X ) = ( Base ` X ) )
30 29 fveq2d 
 |-  ( ( ph /\ X e. B ) -> ( ( Id ` S ) ` ( F ` X ) ) = ( ( Id ` S ) ` ( Base ` X ) ) )
31 eqid 
 |-  ( Id ` S ) = ( Id ` S )
32 1 3 5 estrcbasbas 
 |-  ( ( ph /\ X e. B ) -> ( Base ` X ) e. U )
33 2 31 12 32 setcid 
 |-  ( ( ph /\ X e. B ) -> ( ( Id ` S ) ` ( Base ` X ) ) = ( _I |` ( Base ` X ) ) )
34 30 33 eqtr2d 
 |-  ( ( ph /\ X e. B ) -> ( _I |` ( Base ` X ) ) = ( ( Id ` S ) ` ( F ` X ) ) )
35 18 28 34 3eqtrd 
 |-  ( ( ph /\ X e. B ) -> ( ( X G X ) ` ( ( Id ` E ) ` X ) ) = ( ( Id ` S ) ` ( F ` X ) ) )