Theorem ghmrn 14037

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 2238	. . . 4 |- ( Base ` S ) = ( Base ` S )
2		eqid 2238	. . . 4 |- ( Base ` T ) = ( Base ` T )
3	1, 2	ghmf 14027	. . 3 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
4	3	frnd 5538	. 2 |- ( F e. ( S GrpHom T ) -> ran F C_ ( Base ` T ) )
5		ghmgrp1 14025	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
6		eqid 2238	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
7	1, 6	grpidcl 13811	. . . . . 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 5535	. . . . 5 |- ( F e. ( S GrpHom T ) -> dom F = ( Base ` S ) )
10	8, 9	eleqtrrd 2318	. . . 4 |- ( F e. ( S GrpHom T ) -> ( 0g ` S ) e. dom F )
11		elex2 2838	. . . 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 4996	. . 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 2238	. . . . . . . . . 10 |- ( +g ` S ) = ( +g ` S )
16		eqid 2238	. . . . . . . . . 10 |- ( +g ` T ) = ( +g ` T )
17	1, 15, 16	ghmlin 14028	. . . . . . . . 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 5529	. . . . . . . . . . 11 |- ( F e. ( S GrpHom T ) -> F Fn ( Base ` S ) )
19	18	3ad2ant1 1049	. . . . . . . . . 10 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> F Fn ( Base ` S ) )
20	1, 15	grpcl 13790	. . . . . . . . . . 11 |- ( ( S e. Grp /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> ( c ( +g ` S ) a ) e. ( Base ` S ) )
21	5, 20	syl3an1 1311	. . . . . . . . . 10 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> ( c ( +g ` S ) a ) e. ( Base ` S ) )
22		fnfvelrn 5831	. . . . . . . . . 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 415	. . . . . . . . 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 2316	. . . . . . . 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 1236	. . . . . . 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 2622	. . . . . 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 6083	. . . . . . . . . 10 |- ( b = ( F ` a ) -> ( ( F ` c ) ( +g ` T ) b ) = ( ( F ` c ) ( +g ` T ) ( F ` a ) ) )
28	27	eleq1d 2307	. . . . . . . . 9 |- ( b = ( F ` a ) -> ( ( ( F ` c ) ( +g ` T ) b ) e. ran F <-> ( ( F ` c ) ( +g ` T ) ( F ` a ) ) e. ran F ) )
29	28	ralrn 5837	. . . . . . . 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 2238	. . . . . . 7 |- ( invg ` S ) = ( invg ` S )
34		eqid 2238	. . . . . . 7 |- ( invg ` T ) = ( invg ` T )
35	1, 33, 34	ghminv 14030	. . . . . 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 13830	. . . . . . . 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 5831	. . . . . . 7 |- ( ( F Fn ( Base ` S ) /\ ( ( invg ` S ) ` c ) e. ( Base ` S ) ) -> ( F ` ( ( invg ` S ) ` c ) ) e. ran F )
40	36, 38, 39	syl2anc 415	. . . . . 6 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) ) -> ( F ` ( ( invg ` S ) ` c ) ) e. ran F )
41	35, 40	eqeltrrd 2316	. . . . 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 2623	. . 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 6082	. . . . . . . 8 |- ( a = ( F ` c ) -> ( a ( +g ` T ) b ) = ( ( F ` c ) ( +g ` T ) b ) )
45	44	eleq1d 2307	. . . . . . 7 |- ( a = ( F ` c ) -> ( ( a ( +g ` T ) b ) e. ran F <-> ( ( F ` c ) ( +g ` T ) b ) e. ran F ) )
46	45	ralbidv 2550	. . . . . 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 5690	. . . . . . 7 |- ( a = ( F ` c ) -> ( ( invg ` T ) ` a ) = ( ( invg ` T ) ` ( F ` c ) ) )
48	47	eleq1d 2307	. . . . . 6 |- ( a = ( F ` c ) -> ( ( ( invg ` T ) ` a ) e. ran F <-> ( ( invg ` T ) ` ( F ` c ) ) e. ran F ) )
49	46, 48	anbi12d 477	. . . . 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 5837	. . . 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 14026	. . 3 |- ( F e. ( S GrpHom T ) -> T e. Grp )
54	2, 16, 34	issubg2m 13969	. . 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 1211	1 |- ( F e. ( S GrpHom T ) -> ran F e. (SubGrp ` T ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   C_ wss 3220   dom cdm 4769   ran crn 4770   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   0gc0g 13587   Grpcgrp 13782   invgcminusg 13783  SubGrpcsubg 13947   GrpHom cghm 14020

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-subg 13950  df-ghm 14021

This theorem is referenced by:  ghmghmrn  14043  ghmima  14045