Theorem ghmrn 13387

Description: The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Assertion

Ref	Expression
ghmrn	|- ( F e. ( S GrpHom T ) -> ran F e. (SubGrp ` T ) )

Proof of Theorem ghmrn

Dummy variables a b c j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2196	. . . 4 |- ( Base ` S ) = ( Base ` S )
2		eqid 2196	. . . 4 |- ( Base ` T ) = ( Base ` T )
3	1, 2	ghmf 13377	. . 3 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
4	3	frnd 5417	. 2 |- ( F e. ( S GrpHom T ) -> ran F C_ ( Base ` T ) )
5		ghmgrp1 13375	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
6		eqid 2196	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
7	1, 6	grpidcl 13161	. . . . . 6 |- ( S e. Grp -> ( 0g ` S ) e. ( Base ` S ) )
8	5, 7	syl 14	. . . . 5 |- ( F e. ( S GrpHom T ) -> ( 0g ` S ) e. ( Base ` S ) )
9	3	fdmd 5414	. . . . 5 |- ( F e. ( S GrpHom T ) -> dom F = ( Base ` S ) )
10	8, 9	eleqtrrd 2276	. . . 4 |- ( F e. ( S GrpHom T ) -> ( 0g ` S ) e. dom F )
11		elex2 2779	. . . 4 |- ( ( 0g ` S ) e. dom F -> E. j j e. dom F )
12	10, 11	syl 14	. . 3 |- ( F e. ( S GrpHom T ) -> E. j j e. dom F )
13		dmmrnm 4885	. . 3 |- ( E. j j e. dom F <-> E. j j e. ran F )
14	12, 13	sylib 122	. 2 |- ( F e. ( S GrpHom T ) -> E. j j e. ran F )
15		eqid 2196	. . . . . . . . . 10 |- ( +g ` S ) = ( +g ` S )
16		eqid 2196	. . . . . . . . . 10 |- ( +g ` T ) = ( +g ` T )
17	1, 15, 16	ghmlin 13378	. . . . . . . . 9 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> ( F ` ( c ( +g ` S ) a ) ) = ( ( F ` c ) ( +g ` T ) ( F ` a ) ) )
18	3	ffnd 5408	. . . . . . . . . . 11 |- ( F e. ( S GrpHom T ) -> F Fn ( Base ` S ) )
19	18	3ad2ant1 1020	. . . . . . . . . 10 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> F Fn ( Base ` S ) )
20	1, 15	grpcl 13140	. . . . . . . . . . 11 |- ( ( S e. Grp /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> ( c ( +g ` S ) a ) e. ( Base ` S ) )
21	5, 20	syl3an1 1282	. . . . . . . . . 10 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> ( c ( +g ` S ) a ) e. ( Base ` S ) )
22		fnfvelrn 5694	. . . . . . . . . 10 |- ( ( F Fn ( Base ` S ) /\ ( c ( +g ` S ) a ) e. ( Base ` S ) ) -> ( F ` ( c ( +g ` S ) a ) ) e. ran F )
23	19, 21, 22	syl2anc 411	. . . . . . . . 9 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> ( F ` ( c ( +g ` S ) a ) ) e. ran F )
24	17, 23	eqeltrrd 2274	. . . . . . . 8 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> ( ( F ` c ) ( +g ` T ) ( F ` a ) ) e. ran F )
25	24	3expia 1207	. . . . . . 7 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) ) -> ( a e. ( Base ` S ) -> ( ( F ` c ) ( +g ` T ) ( F ` a ) ) e. ran F ) )
26	25	ralrimiv 2569	. . . . . 6 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) ) -> A. a e. ( Base ` S ) ( ( F ` c ) ( +g ` T ) ( F ` a ) ) e. ran F )
27		oveq2 5930	. . . . . . . . . 10 |- ( b = ( F ` a ) -> ( ( F ` c ) ( +g ` T ) b ) = ( ( F ` c ) ( +g ` T ) ( F ` a ) ) )
28	27	eleq1d 2265	. . . . . . . . 9 |- ( b = ( F ` a ) -> ( ( ( F ` c ) ( +g ` T ) b ) e. ran F <-> ( ( F ` c ) ( +g ` T ) ( F ` a ) ) e. ran F ) )
29	28	ralrn 5700	. . . . . . . 8 |- ( F Fn ( Base ` S ) -> ( A. b e. ran F ( ( F ` c ) ( +g ` T ) b ) e. ran F <-> A. a e. ( Base ` S ) ( ( F ` c ) ( +g ` T ) ( F ` a ) ) e. ran F ) )
30	18, 29	syl 14	. . . . . . 7 |- ( F e. ( S GrpHom T ) -> ( A. b e. ran F ( ( F ` c ) ( +g ` T ) b ) e. ran F <-> A. a e. ( Base ` S ) ( ( F ` c ) ( +g ` T ) ( F ` a ) ) e. ran F ) )
31	30	adantr 276	. . . . . 6 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) ) -> ( A. b e. ran F ( ( F ` c ) ( +g ` T ) b ) e. ran F <-> A. a e. ( Base ` S ) ( ( F ` c ) ( +g ` T ) ( F ` a ) ) e. ran F ) )
32	26, 31	mpbird 167	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) ) -> A. b e. ran F ( ( F ` c ) ( +g ` T ) b ) e. ran F )
33		eqid 2196	. . . . . . 7 |- ( invg ` S ) = ( invg ` S )
34		eqid 2196	. . . . . . 7 |- ( invg ` T ) = ( invg ` T )
35	1, 33, 34	ghminv 13380	. . . . . 6 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) ) -> ( F ` ( ( invg ` S ) ` c ) ) = ( ( invg ` T ) ` ( F ` c ) ) )
36	18	adantr 276	. . . . . . 7 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) ) -> F Fn ( Base ` S ) )
37	1, 33	grpinvcl 13180	. . . . . . . 8 |- ( ( S e. Grp /\ c e. ( Base ` S ) ) -> ( ( invg ` S ) ` c ) e. ( Base ` S ) )
38	5, 37	sylan 283	. . . . . . 7 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) ) -> ( ( invg ` S ) ` c ) e. ( Base ` S ) )
39		fnfvelrn 5694	. . . . . . 7 |- ( ( F Fn ( Base ` S ) /\ ( ( invg ` S ) ` c ) e. ( Base ` S ) ) -> ( F ` ( ( invg ` S ) ` c ) ) e. ran F )
40	36, 38, 39	syl2anc 411	. . . . . 6 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) ) -> ( F ` ( ( invg ` S ) ` c ) ) e. ran F )
41	35, 40	eqeltrrd 2274	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) ) -> ( ( invg ` T ) ` ( F ` c ) ) e. ran F )
42	32, 41	jca 306	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) ) -> ( A. b e. ran F ( ( F ` c ) ( +g ` T ) b ) e. ran F /\ ( ( invg ` T ) ` ( F ` c ) ) e. ran F ) )
43	42	ralrimiva 2570	. . 3 |- ( F e. ( S GrpHom T ) -> A. c e. ( Base ` S ) ( A. b e. ran F ( ( F ` c ) ( +g ` T ) b ) e. ran F /\ ( ( invg ` T ) ` ( F ` c ) ) e. ran F ) )
44		oveq1 5929	. . . . . . . 8 |- ( a = ( F ` c ) -> ( a ( +g ` T ) b ) = ( ( F ` c ) ( +g ` T ) b ) )
45	44	eleq1d 2265	. . . . . . 7 |- ( a = ( F ` c ) -> ( ( a ( +g ` T ) b ) e. ran F <-> ( ( F ` c ) ( +g ` T ) b ) e. ran F ) )
46	45	ralbidv 2497	. . . . . 6 |- ( a = ( F ` c ) -> ( A. b e. ran F ( a ( +g ` T ) b ) e. ran F <-> A. b e. ran F ( ( F ` c ) ( +g ` T ) b ) e. ran F ) )
47		fveq2 5558	. . . . . . 7 |- ( a = ( F ` c ) -> ( ( invg ` T ) ` a ) = ( ( invg ` T ) ` ( F ` c ) ) )
48	47	eleq1d 2265	. . . . . 6 |- ( a = ( F ` c ) -> ( ( ( invg ` T ) ` a ) e. ran F <-> ( ( invg ` T ) ` ( F ` c ) ) e. ran F ) )
49	46, 48	anbi12d 473	. . . . 5 |- ( a = ( F ` c ) -> ( ( A. b e. ran F ( a ( +g ` T ) b ) e. ran F /\ ( ( invg ` T ) ` a ) e. ran F ) <-> ( A. b e. ran F ( ( F ` c ) ( +g ` T ) b ) e. ran F /\ ( ( invg ` T ) ` ( F ` c ) ) e. ran F ) ) )
50	49	ralrn 5700	. . . 4 |- ( F Fn ( Base ` S ) -> ( A. a e. ran F ( A. b e. ran F ( a ( +g ` T ) b ) e. ran F /\ ( ( invg ` T ) ` a ) e. ran F ) <-> A. c e. ( Base ` S ) ( A. b e. ran F ( ( F ` c ) ( +g ` T ) b ) e. ran F /\ ( ( invg ` T ) ` ( F ` c ) ) e. ran F ) ) )
51	18, 50	syl 14	. . 3 |- ( F e. ( S GrpHom T ) -> ( A. a e. ran F ( A. b e. ran F ( a ( +g ` T ) b ) e. ran F /\ ( ( invg ` T ) ` a ) e. ran F ) <-> A. c e. ( Base ` S ) ( A. b e. ran F ( ( F ` c ) ( +g ` T ) b ) e. ran F /\ ( ( invg ` T ) ` ( F ` c ) ) e. ran F ) ) )
52	43, 51	mpbird 167	. 2 |- ( F e. ( S GrpHom T ) -> A. a e. ran F ( A. b e. ran F ( a ( +g ` T ) b ) e. ran F /\ ( ( invg ` T ) ` a ) e. ran F ) )
53		ghmgrp2 13376	. . 3 |- ( F e. ( S GrpHom T ) -> T e. Grp )
54	2, 16, 34	issubg2m 13319	. . 3 |- ( T e. Grp -> ( ran F e. (SubGrp ` T ) <-> ( ran F C_ ( Base ` T ) /\ E. j j e. ran F /\ A. a e. ran F ( A. b e. ran F ( a ( +g ` T ) b ) e. ran F /\ ( ( invg ` T ) ` a ) e. ran F ) ) ) )
55	53, 54	syl 14	. 2 |- ( F e. ( S GrpHom T ) -> ( ran F e. (SubGrp ` T ) <-> ( ran F C_ ( Base ` T ) /\ E. j j e. ran F /\ A. a e. ran F ( A. b e. ran F ( a ( +g ` T ) b ) e. ran F /\ ( ( invg ` T ) ` a ) e. ran F ) ) ) )
56	4, 14, 52, 55	mpbir3and 1182	1 |- ( F e. ( S GrpHom T ) -> ran F e. (SubGrp ` T ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1506   e. wcel 2167   A.wral 2475   C_ wss 3157   dom cdm 4663   ran crn 4664   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Grpcgrp 13132   invgcminusg 13133  SubGrpcsubg 13297   GrpHom cghm 13370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300  df-ghm 13371

This theorem is referenced by:  ghmghmrn  13393  ghmima  13395