Theorem elghomlem1OLD 33684

Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 33686. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
elghomlem1OLD.1	|- S = { f | ( f : ran G --> ran H /\ A. x e. ran G A. y e. ran G ( ( f ` x ) H ( f ` y ) ) = ( f ` ( x G y ) ) ) }

Assertion

Ref	Expression
elghomlem1OLD	|- ( ( G e. GrpOp /\ H e. GrpOp ) -> ( G GrpOpHom H ) = S )

Distinct variable groups:   x, f, y, G   f, H, x, y

Allowed substitution hints:   S( x, y, f)

Proof of Theorem elghomlem1OLD

Dummy variables g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnexg 7098	. . 3 |- ( G e. GrpOp -> ran G e. _V )
2		rnexg 7098	. . 3 |- ( H e. GrpOp -> ran H e. _V )
3		elghomlem1OLD.1	. . . 4 |- S = { f | ( f : ran G --> ran H /\ A. x e. ran G A. y e. ran G ( ( f ` x ) H ( f ` y ) ) = ( f ` ( x G y ) ) ) }
4	3	fabexg 7122	. . 3 |- ( ( ran G e. _V /\ ran H e. _V ) -> S e. _V )
5	1, 2, 4	syl2an 494	. 2 |- ( ( G e. GrpOp /\ H e. GrpOp ) -> S e. _V )
6		rneq 5351	. . . . . 6 |- ( g = G -> ran g = ran G )
7	6	feq2d 6031	. . . . 5 |- ( g = G -> ( f : ran g --> ran h <-> f : ran G --> ran h ) )
8		oveq 6656	. . . . . . . . 9 |- ( g = G -> ( x g y ) = ( x G y ) )
9	8	fveq2d 6195	. . . . . . . 8 |- ( g = G -> ( f ` ( x g y ) ) = ( f ` ( x G y ) ) )
10	9	eqeq2d 2632	. . . . . . 7 |- ( g = G -> ( ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x g y ) ) <-> ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x G y ) ) ) )
11	6, 10	raleqbidv 3152	. . . . . 6 |- ( g = G -> ( A. y e. ran g ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x g y ) ) <-> A. y e. ran G ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x G y ) ) ) )
12	6, 11	raleqbidv 3152	. . . . 5 |- ( g = G -> ( A. x e. ran g A. y e. ran g ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x g y ) ) <-> A. x e. ran G A. y e. ran G ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x G y ) ) ) )
13	7, 12	anbi12d 747	. . . 4 |- ( g = G -> ( ( f : ran g --> ran h /\ A. x e. ran g A. y e. ran g ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x g y ) ) ) <-> ( f : ran G --> ran h /\ A. x e. ran G A. y e. ran G ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x G y ) ) ) ) )
14	13	abbidv 2741	. . 3 |- ( g = G -> { f | ( f : ran g --> ran h /\ A. x e. ran g A. y e. ran g ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x g y ) ) ) } = { f | ( f : ran G --> ran h /\ A. x e. ran G A. y e. ran G ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x G y ) ) ) } )
15		rneq 5351	. . . . . . 7 |- ( h = H -> ran h = ran H )
16	15	feq3d 6032	. . . . . 6 |- ( h = H -> ( f : ran G --> ran h <-> f : ran G --> ran H ) )
17		oveq 6656	. . . . . . . 8 |- ( h = H -> ( ( f ` x ) h ( f ` y ) ) = ( ( f ` x ) H ( f ` y ) ) )
18	17	eqeq1d 2624	. . . . . . 7 |- ( h = H -> ( ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x G y ) ) <-> ( ( f ` x ) H ( f ` y ) ) = ( f ` ( x G y ) ) ) )
19	18	2ralbidv 2989	. . . . . 6 |- ( h = H -> ( A. x e. ran G A. y e. ran G ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x G y ) ) <-> A. x e. ran G A. y e. ran G ( ( f ` x ) H ( f ` y ) ) = ( f ` ( x G y ) ) ) )
20	16, 19	anbi12d 747	. . . . 5 |- ( h = H -> ( ( f : ran G --> ran h /\ A. x e. ran G A. y e. ran G ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x G y ) ) ) <-> ( f : ran G --> ran H /\ A. x e. ran G A. y e. ran G ( ( f ` x ) H ( f ` y ) ) = ( f ` ( x G y ) ) ) ) )
21	20	abbidv 2741	. . . 4 |- ( h = H -> { f | ( f : ran G --> ran h /\ A. x e. ran G A. y e. ran G ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x G y ) ) ) } = { f | ( f : ran G --> ran H /\ A. x e. ran G A. y e. ran G ( ( f ` x ) H ( f ` y ) ) = ( f ` ( x G y ) ) ) } )
22	21, 3	syl6eqr 2674	. . 3 |- ( h = H -> { f | ( f : ran G --> ran h /\ A. x e. ran G A. y e. ran G ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x G y ) ) ) } = S )
23		df-ghomOLD 33683	. . 3 |- GrpOpHom = ( g e. GrpOp , h e. GrpOp |-> { f | ( f : ran g --> ran h /\ A. x e. ran g A. y e. ran g ( ( f ` x ) h ( f ` y ) ) = ( f ` ( x g y ) ) ) } )
24	14, 22, 23	ovmpt2g 6795	. 2 |- ( ( G e. GrpOp /\ H e. GrpOp /\ S e. _V ) -> ( G GrpOpHom H ) = S )
25	5, 24	mpd3an3 1425	1 |- ( ( G e. GrpOp /\ H e. GrpOp ) -> ( G GrpOpHom H ) = S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   GrpOpcgr 27343   GrpOpHom cghomOLD 33682

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-ghomOLD 33683

This theorem is referenced by:  elghomlem2OLD  33685