Theorem ghomidOLD 33688

Description: Obsolete version of ghmid 17666 as of 15-Mar-2020. A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
ghomidOLD.1	|- U = (GId ` G )
ghomidOLD.2	|- T = (GId ` H )

Assertion

Ref	Expression
ghomidOLD	|- ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) -> ( F ` U ) = T )

Proof of Theorem ghomidOLD

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . 7 |- ran G = ran G
2		ghomidOLD.1	. . . . . . 7 |- U = (GId ` G )
3	1, 2	grpoidcl 27368	. . . . . 6 |- ( G e. GrpOp -> U e. ran G )
4	3	3ad2ant1 1082	. . . . 5 |- ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) -> U e. ran G )
5	4, 4	jca 554	. . . 4 |- ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) -> ( U e. ran G /\ U e. ran G ) )
6	1	ghomlinOLD 33687	. . . 4 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( U e. ran G /\ U e. ran G ) ) -> ( ( F ` U ) H ( F ` U ) ) = ( F ` ( U G U ) ) )
7	5, 6	mpdan 702	. . 3 |- ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) -> ( ( F ` U ) H ( F ` U ) ) = ( F ` ( U G U ) ) )
8	1, 2	grpolid 27370	. . . . . 6 |- ( ( G e. GrpOp /\ U e. ran G ) -> ( U G U ) = U )
9	3, 8	mpdan 702	. . . . 5 |- ( G e. GrpOp -> ( U G U ) = U )
10	9	fveq2d 6195	. . . 4 |- ( G e. GrpOp -> ( F ` ( U G U ) ) = ( F ` U ) )
11	10	3ad2ant1 1082	. . 3 |- ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) -> ( F ` ( U G U ) ) = ( F ` U ) )
12	7, 11	eqtrd 2656	. 2 |- ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) -> ( ( F ` U ) H ( F ` U ) ) = ( F ` U ) )
13		eqid 2622	. . . . . . 7 |- ran H = ran H
14	1, 13	elghomOLD 33686	. . . . . 6 |- ( ( 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 ) ) ) ) )
15	14	biimp3a 1432	. . . . 5 |- ( ( 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 ) ) ) )
16	15	simpld 475	. . . 4 |- ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) -> F : ran G --> ran H )
17	16, 4	ffvelrnd 6360	. . 3 |- ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) -> ( F ` U ) e. ran H )
18		ghomidOLD.2	. . . . . 6 |- T = (GId ` H )
19	13, 18	grpoid 27374	. . . . 5 |- ( ( H e. GrpOp /\ ( F ` U ) e. ran H ) -> ( ( F ` U ) = T <-> ( ( F ` U ) H ( F ` U ) ) = ( F ` U ) ) )
20	19	ex 450	. . . 4 |- ( H e. GrpOp -> ( ( F ` U ) e. ran H -> ( ( F ` U ) = T <-> ( ( F ` U ) H ( F ` U ) ) = ( F ` U ) ) ) )
21	20	3ad2ant2 1083	. . 3 |- ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) -> ( ( F ` U ) e. ran H -> ( ( F ` U ) = T <-> ( ( F ` U ) H ( F ` U ) ) = ( F ` U ) ) ) )
22	17, 21	mpd 15	. 2 |- ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) -> ( ( F ` U ) = T <-> ( ( F ` U ) H ( F ` U ) ) = ( F ` U ) ) )
23	12, 22	mpbird 247	1 |- ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) -> ( F ` U ) = T )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   GrpOpcgr 27343  GIdcgi 27344   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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-grpo 27347  df-gid 27348  df-ghomOLD 33683

This theorem is referenced by:  grpokerinj  33692  rngohom0  33771