Theorem ghmpreima 14046

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 5145	. . 3 |- ( `' F " V ) C_ dom F
2		eqid 2238	. . . . 5 |- ( Base ` S ) = ( Base ` S )
3		eqid 2238	. . . . 5 |- ( Base ` T ) = ( Base ` T )
4	2, 3	ghmf 14027	. . . 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 5544	. 2 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( `' F " V ) C_ ( Base ` S ) )
7		ghmgrp1 14025	. . . . . 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 2238	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
10	2, 9	grpidcl 13811	. . . . 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 2238	. . . . . . 7 |- ( 0g ` T ) = ( 0g ` T )
13	9, 12	ghmid 14029	. . . . . 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 13962	. . . . . 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 2315	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( F ` ( 0g ` S ) ) e. V )
18	5	ffnd 5529	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> F Fn ( Base ` S ) )
19		elpreima 5819	. . . . 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 957	. . 3 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( 0g ` S ) e. ( `' F " V ) )
22		elex2 2838	. . 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 5819	. . . . 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 5819	. . . . . . . . . 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 2238	. . . . . . . . . . 11 |- ( +g ` S ) = ( +g ` S )
30	7	ad2antrr 492	. . . . . . . . . . 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 543	. . . . . . . . . . 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 545	. . . . . . . . . . 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 13796	. . . . . . . . . 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 531	. . . . . . . . . . . 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 2238	. . . . . . . . . . . . 13 |- ( +g ` T ) = ( +g ` T )
36	2, 29, 35	ghmlin 14028	. . . . . . . . . . . 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 1278	. . . . . . . . . . 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 533	. . . . . . . . . . . 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 544	. . . . . . . . . . . 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 546	. . . . . . . . . . . 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 13964	. . . . . . . . . . . 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 1278	. . . . . . . . . . 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 2315	. . . . . . . . . 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 5819	. . . . . . . . . . . 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 957	. . . . . . . . 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 2622	. . . . . 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 535	. . . . . . . 8 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> a e. ( Base ` S ) )
52		eqid 2238	. . . . . . . . 9 |- ( invg ` S ) = ( invg ` S )
53	2, 52	grpinvcl 13830	. . . . . . . 8 |- ( ( S e. Grp /\ a e. ( Base ` S ) ) -> ( ( invg ` S ) ` a ) e. ( Base ` S ) )
54	8, 51, 53	syl2an2r 603	. . . . . . 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 2238	. . . . . . . . . 10 |- ( invg ` T ) = ( invg ` T )
56	2, 52, 55	ghminv 14030	. . . . . . . . 9 |- ( ( F e. ( S GrpHom T ) /\ a e. ( Base ` S ) ) -> ( F ` ( ( invg ` S ) ` a ) ) = ( ( invg ` T ) ` ( F ` a ) ) )
57	56	ad2ant2r 513	. . . . . . . 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 13963	. . . . . . . . 9 |- ( ( V e. (SubGrp ` T ) /\ ( F ` a ) e. V ) -> ( ( invg ` T ) ` ( F ` a ) ) e. V )
59	58	ad2ant2l 512	. . . . . . . 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 2315	. . . . . . 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 5819	. . . . . . . . 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 957	. . . . . 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 2622	. 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 13969	. . 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 1211	1 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( `' F " V ) e. (SubGrp ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   C_ wss 3220   `'ccnv 4768   "cima 4772   Fn wfn 5367   -->wf 5368   ` 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:  ghmnsgpreima  14049