Theorem ghmrn 14010

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 2234	. . . 4 |- ( Base ` S ) = ( Base ` S )
2		eqid 2234	. . . 4 |- ( Base ` T ) = ( Base ` T )
3	1, 2	ghmf 14000	. . 3 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
4	3	frnd 5523	. 2 |- ( F e. ( S GrpHom T ) -> ran F C_ ( Base ` T ) )
5		ghmgrp1 13998	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
6		eqid 2234	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
7	1, 6	grpidcl 13784	. . . . . 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 5520	. . . . 5 |- ( F e. ( S GrpHom T ) -> dom F = ( Base ` S ) )
10	8, 9	eleqtrrd 2314	. . . 4 |- ( F e. ( S GrpHom T ) -> ( 0g ` S ) e. dom F )
11		elex2 2832	. . . 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 4981	. . 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 2234	. . . . . . . . . 10 |- ( +g ` S ) = ( +g ` S )
16		eqid 2234	. . . . . . . . . 10 |- ( +g ` T ) = ( +g ` T )
17	1, 15, 16	ghmlin 14001	. . . . . . . . 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 5514	. . . . . . . . . . 11 |- ( F e. ( S GrpHom T ) -> F Fn ( Base ` S ) )
19	18	3ad2ant1 1045	. . . . . . . . . 10 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> F Fn ( Base ` S ) )
20	1, 15	grpcl 13763	. . . . . . . . . . 11 |- ( ( S e. Grp /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> ( c ( +g ` S ) a ) e. ( Base ` S ) )
21	5, 20	syl3an1 1307	. . . . . . . . . 10 |- ( ( F e. ( S GrpHom T ) /\ c e. ( Base ` S ) /\ a e. ( Base ` S ) ) -> ( c ( +g ` S ) a ) e. ( Base ` S ) )
22		fnfvelrn 5814	. . . . . . . . . 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 2312	. . . . . . . 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 1232	. . . . . . 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 2616	. . . . . 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 6066	. . . . . . . . . 10 |- ( b = ( F ` a ) -> ( ( F ` c ) ( +g ` T ) b ) = ( ( F ` c ) ( +g ` T ) ( F ` a ) ) )
28	27	eleq1d 2303	. . . . . . . . 9 |- ( b = ( F ` a ) -> ( ( ( F ` c ) ( +g ` T ) b ) e. ran F <-> ( ( F ` c ) ( +g ` T ) ( F ` a ) ) e. ran F ) )
29	28	ralrn 5820	. . . . . . . 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 2234	. . . . . . 7 |- ( invg ` S ) = ( invg ` S )
34		eqid 2234	. . . . . . 7 |- ( invg ` T ) = ( invg ` T )
35	1, 33, 34	ghminv 14003	. . . . . 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 13803	. . . . . . . 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 5814	. . . . . . 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 2312	. . . . 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 2617	. . 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 6065	. . . . . . . 8 |- ( a = ( F ` c ) -> ( a ( +g ` T ) b ) = ( ( F ` c ) ( +g ` T ) b ) )
45	44	eleq1d 2303	. . . . . . 7 |- ( a = ( F ` c ) -> ( ( a ( +g ` T ) b ) e. ran F <-> ( ( F ` c ) ( +g ` T ) b ) e. ran F ) )
46	45	ralbidv 2544	. . . . . 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 5675	. . . . . . 7 |- ( a = ( F ` c ) -> ( ( invg ` T ) ` a ) = ( ( invg ` T ) ` ( F ` c ) ) )
48	47	eleq1d 2303	. . . . . 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 5820	. . . 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 13999	. . 3 |- ( F e. ( S GrpHom T ) -> T e. Grp )
54	2, 16, 34	issubg2m 13942	. . 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 1207	1 |- ( F e. ( S GrpHom T ) -> ran F e. (SubGrp ` T ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2205   A.wral 2522   C_ wss 3214   dom cdm 4754   ran crn 4755   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   0gc0g 13553   Grpcgrp 13755   invgcminusg 13756  SubGrpcsubg 13920   GrpHom cghm 13993

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-subg 13923  df-ghm 13994

This theorem is referenced by:  ghmghmrn  14016  ghmima  14018