Theorem ghmpreima 13852

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 5099	. . 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 13833	. . . 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 5497	. 2 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( `' F " V ) C_ ( Base ` S ) )
7		ghmgrp1 13831	. . . . . 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 13611	. . . . 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 13835	. . . . . 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 13768	. . . . . 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 5483	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> F Fn ( Base ` S ) )
19		elpreima 5766	. . . . 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 952	. . 3 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( 0g ` S ) e. ( `' F " V ) )
22		elex2 2819	. . 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 5766	. . . . 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 5766	. . . . . . . . . 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 13596	. . . . . . . . . 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 13834	. . . . . . . . . . . 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 1273	. . . . . . . . . . 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 13770	. . . . . . . . . . . 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 1273	. . . . . . . . . . 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 5766	. . . . . . . . . . . 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 952	. . . . . . . . 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 2604	. . . . . 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 13630	. . . . . . . 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 13836	. . . . . . . . 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 13769	. . . . . . . . 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 5766	. . . . . . . . 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 952	. . . . . 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 2604	. 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 13775	. . 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 1206	1 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( `' F " V ) e. (SubGrp ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   C_ wss 3200   `'ccnv 4724   "cima 4728   Fn wfn 5321   -->wf 5322   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338   Grpcgrp 13582   invgcminusg 13583  SubGrpcsubg 13753   GrpHom cghm 13826

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-subg 13756  df-ghm 13827

This theorem is referenced by:  ghmnsgpreima  13855