Theorem funcestrcsetclem6 16785

Description: Lemma 6 for funcestrcsetc 16789. (Contributed by AV, 23-Mar-2020.)

Hypotheses

Ref	Expression
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 ) ) ) ) )
funcestrcsetc.m	|- M = ( Base ` X )
funcestrcsetc.n	|- N = ( Base ` Y )

Assertion

Ref	Expression
funcestrcsetclem6	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( N ^m M ) ) -> ( ( X G Y ) ` H ) = H )

Distinct variable groups:   x, B   x, X   ph, x   x, C   y, B, x   y, X   ph, y   x, M, y   x, N, y   x, Y, y

Allowed substitution hints:   C( y)   S( x, y)   U( x, y)   E( x, y)   F( x, y)   G( x, y)   H( x, y)

Proof of Theorem funcestrcsetclem6

Step	Hyp	Ref	Expression
1		funcestrcsetc.e	. . . . 5 |- E = (ExtStrCat ` U )
2		funcestrcsetc.s	. . . . 5 |- S = ( SetCat ` U )
3		funcestrcsetc.b	. . . . 5 |- B = ( Base ` E )
4		funcestrcsetc.c	. . . . 5 |- C = ( Base ` S )
5		funcestrcsetc.u	. . . . 5 |- ( ph -> U e. WUni )
6		funcestrcsetc.f	. . . . 5 |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
7		funcestrcsetc.g	. . . . 5 |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )
8		funcestrcsetc.m	. . . . 5 |- M = ( Base ` X )
9		funcestrcsetc.n	. . . . 5 |- N = ( Base ` Y )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	funcestrcsetclem5 16784	. . . 4 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X G Y ) = ( _I |` ( N ^m M ) ) )
11	10	3adant3 1081	. . 3 |- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( N ^m M ) ) -> ( X G Y ) = ( _I |` ( N ^m M ) ) )
12	11	fveq1d 6193	. 2 |- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( N ^m M ) ) -> ( ( X G Y ) ` H ) = ( ( _I |` ( N ^m M ) ) ` H ) )
13		fvresi 6439	. . 3 |- ( H e. ( N ^m M ) -> ( ( _I |` ( N ^m M ) ) ` H ) = H )
14	13	3ad2ant3 1084	. 2 |- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( N ^m M ) ) -> ( ( _I |` ( N ^m M ) ) ` H ) = H )
15	12, 14	eqtrd 2656	1 |- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( N ^m M ) ) -> ( ( X G Y ) ` H ) = H )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   |-> cmpt 4729   _I cid 5023   |` cres 5116   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857  WUnicwun 9522   Basecbs 15857   SetCatcsetc 16725  ExtStrCatcestrc 16762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  funcestrcsetclem9  16788  fthestrcsetc  16790  fullestrcsetc  16791