Theorem elghomlem2OLD 33685

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
elghomlem2OLD	|- ( ( G e. GrpOp /\ H e. GrpOp ) -> ( F e. ( G GrpOpHom 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 ) ) ) ) )

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

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

Proof of Theorem elghomlem2OLD

Step	Hyp	Ref	Expression
1		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 ) ) ) }
2	1	elghomlem1OLD 33684	. . 3 |- ( ( G e. GrpOp /\ H e. GrpOp ) -> ( G GrpOpHom H ) = S )
3	2	eleq2d 2687	. 2 |- ( ( G e. GrpOp /\ H e. GrpOp ) -> ( F e. ( G GrpOpHom H ) <-> F e. S ) )
4		elex 3212	. . . . 5 |- ( F e. S -> F e. _V )
5		feq1 6026	. . . . . . . 8 |- ( f = F -> ( f : ran G --> ran H <-> F : ran G --> ran H ) )
6		fveq1 6190	. . . . . . . . . . 11 |- ( f = F -> ( f ` x ) = ( F ` x ) )
7		fveq1 6190	. . . . . . . . . . 11 |- ( f = F -> ( f ` y ) = ( F ` y ) )
8	6, 7	oveq12d 6668	. . . . . . . . . 10 |- ( f = F -> ( ( f ` x ) H ( f ` y ) ) = ( ( F ` x ) H ( F ` y ) ) )
9		fveq1 6190	. . . . . . . . . 10 |- ( f = F -> ( f ` ( x G y ) ) = ( F ` ( x G y ) ) )
10	8, 9	eqeq12d 2637	. . . . . . . . 9 |- ( f = F -> ( ( ( f ` x ) H ( f ` y ) ) = ( f ` ( x G y ) ) <-> ( ( F ` x ) H ( F ` y ) ) = ( F ` ( x G y ) ) ) )
11	10	2ralbidv 2989	. . . . . . . 8 |- ( f = F -> ( 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 ) ) ) )
12	5, 11	anbi12d 747	. . . . . . 7 |- ( f = 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 : ran G --> ran H /\ A. x e. ran G A. y e. ran G ( ( F ` x ) H ( F ` y ) ) = ( F ` ( x G y ) ) ) ) )
13	12, 1	elab2g 3353	. . . . . 6 |- ( F e. _V -> ( F e. S <-> ( 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	biimpd 219	. . . . 5 |- ( F e. _V -> ( F e. S -> ( 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	4, 14	mpcom 38	. . . 4 |- ( F e. S -> ( F : ran G --> ran H /\ A. x e. ran G A. y e. ran G ( ( F ` x ) H ( F ` y ) ) = ( F ` ( x G y ) ) ) )
16		rnexg 7098	. . . . . . 7 |- ( G e. GrpOp -> ran G e. _V )
17		fex 6490	. . . . . . . 8 |- ( ( F : ran G --> ran H /\ ran G e. _V ) -> F e. _V )
18	17	expcom 451	. . . . . . 7 |- ( ran G e. _V -> ( F : ran G --> ran H -> F e. _V ) )
19	16, 18	syl 17	. . . . . 6 |- ( G e. GrpOp -> ( F : ran G --> ran H -> F e. _V ) )
20	19	adantrd 484	. . . . 5 |- ( G e. GrpOp -> ( ( 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 e. _V ) )
21	13	biimprd 238	. . . . 5 |- ( F e. _V -> ( ( 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 e. S ) )
22	20, 21	syli 39	. . . 4 |- ( G e. GrpOp -> ( ( 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 e. S ) )
23	15, 22	impbid2 216	. . 3 |- ( G e. GrpOp -> ( F e. S <-> ( 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	23	adantr 481	. 2 |- ( ( G e. GrpOp /\ H e. GrpOp ) -> ( F e. S <-> ( F : ran G --> ran H /\ A. x e. ran G A. y e. ran G ( ( F ` x ) H ( F ` y ) ) = ( F ` ( x G y ) ) ) ) )
25	3, 24	bitrd 268	1 |- ( ( G e. GrpOp /\ H e. GrpOp ) -> ( F e. ( G GrpOpHom 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 ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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-rep 4771  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-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-pw 4160  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  df-ghomOLD 33683

This theorem is referenced by:  elghomOLD  33686