Theorem funcringcsetcALTV2lem6 42041

Description: Lemma 6 for funcringcsetcALTV2 42045. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
funcringcsetcALTV2.r	|- R = (RingCat ` U )
funcringcsetcALTV2.s	|- S = ( SetCat ` U )
funcringcsetcALTV2.b	|- B = ( Base ` R )
funcringcsetcALTV2.c	|- C = ( Base ` S )
funcringcsetcALTV2.u	|- ( ph -> U e. WUni )
funcringcsetcALTV2.f	|- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
funcringcsetcALTV2.g	|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) )

Assertion

Ref	Expression
funcringcsetcALTV2lem6	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( X RingHom Y ) ) -> ( ( X G Y ) ` H ) = H )

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

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

Proof of Theorem funcringcsetcALTV2lem6

Step	Hyp	Ref	Expression
1		funcringcsetcALTV2.r	. . . . 5 |- R = (RingCat ` U )
2		funcringcsetcALTV2.s	. . . . 5 |- S = ( SetCat ` U )
3		funcringcsetcALTV2.b	. . . . 5 |- B = ( Base ` R )
4		funcringcsetcALTV2.c	. . . . 5 |- C = ( Base ` S )
5		funcringcsetcALTV2.u	. . . . 5 |- ( ph -> U e. WUni )
6		funcringcsetcALTV2.f	. . . . 5 |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
7		funcringcsetcALTV2.g	. . . . 5 |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) )
8	1, 2, 3, 4, 5, 6, 7	funcringcsetcALTV2lem5 42040	. . . 4 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X G Y ) = ( _I |` ( X RingHom Y ) ) )
9	8	3adant3 1081	. . 3 |- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( X RingHom Y ) ) -> ( X G Y ) = ( _I |` ( X RingHom Y ) ) )
10	9	fveq1d 6193	. 2 |- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( X RingHom Y ) ) -> ( ( X G Y ) ` H ) = ( ( _I |` ( X RingHom Y ) ) ` H ) )
11		fvresi 6439	. . 3 |- ( H e. ( X RingHom Y ) -> ( ( _I |` ( X RingHom Y ) ) ` H ) = H )
12	11	3ad2ant3 1084	. 2 |- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( X RingHom Y ) ) -> ( ( _I |` ( X RingHom Y ) ) ` H ) = H )
13	10, 12	eqtrd 2656	1 |- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( X RingHom Y ) ) -> ( ( 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  WUnicwun 9522   Basecbs 15857   SetCatcsetc 16725   RingHom crh 18712  RingCatcringc 42003

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:  funcringcsetcALTV2lem9  42044