Theorem ghmrn 13794

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 2229	. . . 4 |- ( Base ` S ) = ( Base ` S )
2		eqid 2229	. . . 4 |- ( Base ` T ) = ( Base ` T )
3	1, 2	ghmf 13784	. . 3 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
4	3	frnd 5483	. 2 |- ( F e. ( S GrpHom T ) -> ran F C_ ( Base ` T ) )
5		ghmgrp1 13782	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
6		eqid 2229	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
7	1, 6	grpidcl 13562	. . . . . 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 5480	. . . . 5 |- ( F e. ( S GrpHom T ) -> dom F = ( Base ` S ) )
10	8, 9	eleqtrrd 2309	. . . 4 |- ( F e. ( S GrpHom T ) -> ( 0g ` S ) e. dom F )
11		elex2 2816	. . . 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 4943	. . 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 2229	. . . . . . . . . 10 |- ( +g ` S ) = ( +g ` S )
16		eqid 2229	. . . . . . . . . 10 |- ( +g ` T ) = ( +g ` T )
17	1, 15, 16	ghmlin 13785	. . . . . . . . 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 5474	. . . . . . . . . . 11 |- ( F e. ( S GrpHom T ) -> F Fn ( Base ` S ) )
19	18	3ad2ant1 1042	. . . . . . . . . 10 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> F Fn ( Base ` S ) )
20	1, 15	grpcl 13541	. . . . . . . . . . 11 |- ( ( S e. Grp /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> ( c ( +g ` S ) a ) e. ( Base ` S ) )
21	5, 20	syl3an1 1304	. . . . . . . . . 10 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> ( c ( +g ` S ) a ) e. ( Base ` S ) )
22		fnfvelrn 5767	. . . . . . . . . 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 2307	. . . . . . . 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 1229	. . . . . . 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 2602	. . . . . 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 6009	. . . . . . . . . 10 |- ( b = ( F ` a ) -> ( ( F ` c ) ( +g ` T ) b ) = ( ( F ` c ) ( +g ` T ) ( F ` a ) ) )
28	27	eleq1d 2298	. . . . . . . . 9 |- ( b = ( F ` a ) -> ( ( ( F ` c ) ( +g ` T ) b ) e. ran F <-> ( ( F ` c ) ( +g ` T ) ( F ` a ) ) e. ran F ) )
29	28	ralrn 5773	. . . . . . . 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 2229	. . . . . . 7 |- ( invg ` S ) = ( invg ` S )
34		eqid 2229	. . . . . . 7 |- ( invg ` T ) = ( invg ` T )
35	1, 33, 34	ghminv 13787	. . . . . 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 13581	. . . . . . . 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 5767	. . . . . . 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 2307	. . . . 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 2603	. . 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 6008	. . . . . . . 8 |- ( a = ( F ` c ) -> ( a ( +g ` T ) b ) = ( ( F ` c ) ( +g ` T ) b ) )
45	44	eleq1d 2298	. . . . . . 7 |- ( a = ( F ` c ) -> ( ( a ( +g ` T ) b ) e. ran F <-> ( ( F ` c ) ( +g ` T ) b ) e. ran F ) )
46	45	ralbidv 2530	. . . . . 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 5627	. . . . . . 7 |- ( a = ( F ` c ) -> ( ( invg ` T ) ` a ) = ( ( invg ` T ) ` ( F ` c ) ) )
48	47	eleq1d 2298	. . . . . 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 5773	. . . 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 13783	. . 3 |- ( F e. ( S GrpHom T ) -> T e. Grp )
54	2, 16, 34	issubg2m 13726	. . 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 1204	1 |- ( F e. ( S GrpHom T ) -> ran F e. (SubGrp ` T ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   C_ wss 3197   dom cdm 4719   ran crn 4720   Fn wfn 5313   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   0gc0g 13289   Grpcgrp 13533   invgcminusg 13534  SubGrpcsubg 13704   GrpHom cghm 13777

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-iress 13040  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537  df-subg 13707  df-ghm 13778

This theorem is referenced by:  ghmghmrn  13800  ghmima  13802