Theorem ghmpreima 13933

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

Assertion

Ref	Expression
ghmpreima	|- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( `' F " V ) e. (SubGrp ` S ) )

Proof of Theorem ghmpreima

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

Step	Hyp	Ref	Expression
1		cnvimass 5106	. . 3 |- ( `' F " V ) C_ dom F
2		eqid 2231	. . . . 5 |- ( Base ` S ) = ( Base ` S )
3		eqid 2231	. . . . 5 |- ( Base ` T ) = ( Base ` T )
4	2, 3	ghmf 13914	. . . 4 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
5	4	adantr 276	. . 3 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> F : ( Base ` S ) --> ( Base ` T ) )
6	1, 5	fssdm 5504	. 2 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( `' F " V ) C_ ( Base ` S ) )
7		ghmgrp1 13912	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
8	7	adantr 276	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> S e. Grp )
9		eqid 2231	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
10	2, 9	grpidcl 13692	. . . . 5 |- ( S e. Grp -> ( 0g ` S ) e. ( Base ` S ) )
11	8, 10	syl 14	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( 0g ` S ) e. ( Base ` S ) )
12		eqid 2231	. . . . . . 7 |- ( 0g ` T ) = ( 0g ` T )
13	9, 12	ghmid 13916	. . . . . 6 |- ( F e. ( S GrpHom T ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
14	13	adantr 276	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
15	12	subg0cl 13849	. . . . . 6 |- ( V e. (SubGrp ` T ) -> ( 0g ` T ) e. V )
16	15	adantl 277	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( 0g ` T ) e. V )
17	14, 16	eqeltrd 2308	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( F ` ( 0g ` S ) ) e. V )
18	5	ffnd 5490	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> F Fn ( Base ` S ) )
19		elpreima 5775	. . . . 5 |- ( F Fn ( Base ` S ) -> ( ( 0g ` S ) e. ( `' F " V ) <-> ( ( 0g ` S ) e. ( Base ` S ) /\ ( F ` ( 0g ` S ) ) e. V ) ) )
20	18, 19	syl 14	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( ( 0g ` S ) e. ( `' F " V ) <-> ( ( 0g ` S ) e. ( Base ` S ) /\ ( F ` ( 0g ` S ) ) e. V ) ) )
21	11, 17, 20	mpbir2and 953	. . 3 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( 0g ` S ) e. ( `' F " V ) )
22		elex2 2820	. . 3 |- ( ( 0g ` S ) e. ( `' F " V ) -> E. j j e. ( `' F " V ) )
23	21, 22	syl 14	. 2 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> E. j j e. ( `' F " V ) )
24		elpreima 5775	. . . . 5 |- ( F Fn ( Base ` S ) -> ( a e. ( `' F " V ) <-> ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) )
25	18, 24	syl 14	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( a e. ( `' F " V ) <-> ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) )
26		elpreima 5775	. . . . . . . . . 10 |- ( F Fn ( Base ` S ) -> ( b e. ( `' F " V ) <-> ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) )
27	18, 26	syl 14	. . . . . . . . 9 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( b e. ( `' F " V ) <-> ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) )
28	27	adantr 276	. . . . . . . 8 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> ( b e. ( `' F " V ) <-> ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) )
29		eqid 2231	. . . . . . . . . . 11 |- ( +g ` S ) = ( +g ` S )
30	7	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> S e. Grp )
31		simprll 539	. . . . . . . . . . 11 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> a e. ( Base ` S ) )
32		simprrl 541	. . . . . . . . . . 11 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> b e. ( Base ` S ) )
33	2, 29, 30, 31, 32	grpcld 13677	. . . . . . . . . 10 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> ( a ( +g ` S ) b ) e. ( Base ` S ) )
34		simpll 527	. . . . . . . . . . . 12 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> F e. ( S GrpHom T ) )
35		eqid 2231	. . . . . . . . . . . . 13 |- ( +g ` T ) = ( +g ` T )
36	2, 29, 35	ghmlin 13915	. . . . . . . . . . . 12 |- ( ( F e. ( S GrpHom T ) /\ a e. ( Base ` S ) /\ b e. ( Base ` S ) ) -> ( F ` ( a ( +g ` S ) b ) ) = ( ( F ` a ) ( +g ` T ) ( F ` b ) ) )
37	34, 31, 32, 36	syl3anc 1274	. . . . . . . . . . 11 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> ( F ` ( a ( +g ` S ) b ) ) = ( ( F ` a ) ( +g ` T ) ( F ` b ) ) )
38		simplr 529	. . . . . . . . . . . 12 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> V e. (SubGrp ` T ) )
39		simprlr 540	. . . . . . . . . . . 12 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> ( F ` a ) e. V )
40		simprrr 542	. . . . . . . . . . . 12 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> ( F ` b ) e. V )
41	35	subgcl 13851	. . . . . . . . . . . 12 |- ( ( V e. (SubGrp ` T ) /\ ( F ` a ) e. V /\ ( F ` b ) e. V ) -> ( ( F ` a ) ( +g ` T ) ( F ` b ) ) e. V )
42	38, 39, 40, 41	syl3anc 1274	. . . . . . . . . . 11 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> ( ( F ` a ) ( +g ` T ) ( F ` b ) ) e. V )
43	37, 42	eqeltrd 2308	. . . . . . . . . 10 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> ( F ` ( a ( +g ` S ) b ) ) e. V )
44		elpreima 5775	. . . . . . . . . . . 12 |- ( F Fn ( Base ` S ) -> ( ( a ( +g ` S ) b ) e. ( `' F " V ) <-> ( ( a ( +g ` S ) b ) e. ( Base ` S ) /\ ( F ` ( a ( +g ` S ) b ) ) e. V ) ) )
45	18, 44	syl 14	. . . . . . . . . . 11 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( ( a ( +g ` S ) b ) e. ( `' F " V ) <-> ( ( a ( +g ` S ) b ) e. ( Base ` S ) /\ ( F ` ( a ( +g ` S ) b ) ) e. V ) ) )
46	45	adantr 276	. . . . . . . . . 10 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> ( ( a ( +g ` S ) b ) e. ( `' F " V ) <-> ( ( a ( +g ` S ) b ) e. ( Base ` S ) /\ ( F ` ( a ( +g ` S ) b ) ) e. V ) ) )
47	33, 43, 46	mpbir2and 953	. . . . . . . . 9 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) /\ ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) ) ) -> ( a ( +g ` S ) b ) e. ( `' F " V ) )
48	47	expr 375	. . . . . . . 8 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> ( ( b e. ( Base ` S ) /\ ( F ` b ) e. V ) -> ( a ( +g ` S ) b ) e. ( `' F " V ) ) )
49	28, 48	sylbid 150	. . . . . . 7 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> ( b e. ( `' F " V ) -> ( a ( +g ` S ) b ) e. ( `' F " V ) ) )
50	49	ralrimiv 2605	. . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> A. b e. ( `' F " V ) ( a ( +g ` S ) b ) e. ( `' F " V ) )
51		simprl 531	. . . . . . . 8 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> a e. ( Base ` S ) )
52		eqid 2231	. . . . . . . . 9 |- ( invg ` S ) = ( invg ` S )
53	2, 52	grpinvcl 13711	. . . . . . . 8 |- ( ( S e. Grp /\ a e. ( Base ` S ) ) -> ( ( invg ` S ) ` a ) e. ( Base ` S ) )
54	8, 51, 53	syl2an2r 599	. . . . . . 7 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> ( ( invg ` S ) ` a ) e. ( Base ` S ) )
55		eqid 2231	. . . . . . . . . 10 |- ( invg ` T ) = ( invg ` T )
56	2, 52, 55	ghminv 13917	. . . . . . . . 9 |- ( ( F e. ( S GrpHom T ) /\ a e. ( Base ` S ) ) -> ( F ` ( ( invg ` S ) ` a ) ) = ( ( invg ` T ) ` ( F ` a ) ) )
57	56	ad2ant2r 509	. . . . . . . 8 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> ( F ` ( ( invg ` S ) ` a ) ) = ( ( invg ` T ) ` ( F ` a ) ) )
58	55	subginvcl 13850	. . . . . . . . 9 |- ( ( V e. (SubGrp ` T ) /\ ( F ` a ) e. V ) -> ( ( invg ` T ) ` ( F ` a ) ) e. V )
59	58	ad2ant2l 508	. . . . . . . 8 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> ( ( invg ` T ) ` ( F ` a ) ) e. V )
60	57, 59	eqeltrd 2308	. . . . . . 7 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> ( F ` ( ( invg ` S ) ` a ) ) e. V )
61		elpreima 5775	. . . . . . . . 9 |- ( F Fn ( Base ` S ) -> ( ( ( invg ` S ) ` a ) e. ( `' F " V ) <-> ( ( ( invg ` S ) ` a ) e. ( Base ` S ) /\ ( F ` ( ( invg ` S ) ` a ) ) e. V ) ) )
62	18, 61	syl 14	. . . . . . . 8 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( ( ( invg ` S ) ` a ) e. ( `' F " V ) <-> ( ( ( invg ` S ) ` a ) e. ( Base ` S ) /\ ( F ` ( ( invg ` S ) ` a ) ) e. V ) ) )
63	62	adantr 276	. . . . . . 7 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> ( ( ( invg ` S ) ` a ) e. ( `' F " V ) <-> ( ( ( invg ` S ) ` a ) e. ( Base ` S ) /\ ( F ` ( ( invg ` S ) ` a ) ) e. V ) ) )
64	54, 60, 63	mpbir2and 953	. . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> ( ( invg ` S ) ` a ) e. ( `' F " V ) )
65	50, 64	jca 306	. . . . 5 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> ( A. b e. ( `' F " V ) ( a ( +g ` S ) b ) e. ( `' F " V ) /\ ( ( invg ` S ) ` a ) e. ( `' F " V ) ) )
66	65	ex 115	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) -> ( A. b e. ( `' F " V ) ( a ( +g ` S ) b ) e. ( `' F " V ) /\ ( ( invg ` S ) ` a ) e. ( `' F " V ) ) ) )
67	25, 66	sylbid 150	. . 3 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( a e. ( `' F " V ) -> ( A. b e. ( `' F " V ) ( a ( +g ` S ) b ) e. ( `' F " V ) /\ ( ( invg ` S ) ` a ) e. ( `' F " V ) ) ) )
68	67	ralrimiv 2605	. 2 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> A. a e. ( `' F " V ) ( A. b e. ( `' F " V ) ( a ( +g ` S ) b ) e. ( `' F " V ) /\ ( ( invg ` S ) ` a ) e. ( `' F " V ) ) )
69	2, 29, 52	issubg2m 13856	. . 3 |- ( S e. Grp -> ( ( `' F " V ) e. (SubGrp ` S ) <-> ( ( `' F " V ) C_ ( Base ` S ) /\ E. j j e. ( `' F " V ) /\ A. a e. ( `' F " V ) ( A. b e. ( `' F " V ) ( a ( +g ` S ) b ) e. ( `' F " V ) /\ ( ( invg ` S ) ` a ) e. ( `' F " V ) ) ) ) )
70	8, 69	syl 14	. 2 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( ( `' F " V ) e. (SubGrp ` S ) <-> ( ( `' F " V ) C_ ( Base ` S ) /\ E. j j e. ( `' F " V ) /\ A. a e. ( `' F " V ) ( A. b e. ( `' F " V ) ( a ( +g ` S ) b ) e. ( `' F " V ) /\ ( ( invg ` S ) ` a ) e. ( `' F " V ) ) ) ) )
71	6, 23, 68, 70	mpbir3and 1207	1 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( `' F " V ) e. (SubGrp ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2202   A.wral 2511   C_ wss 3201   `'ccnv 4730   "cima 4734   Fn wfn 5328   -->wf 5329   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   0gc0g 13419   Grpcgrp 13663   invgcminusg 13664  SubGrpcsubg 13834   GrpHom cghm 13907

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-subg 13837  df-ghm 13908

This theorem is referenced by:  ghmnsgpreima  13936