Theorem ghmpreima 13717

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 5064	. . 3 |- ( `' F " V ) C_ dom F
2		eqid 2207	. . . . 5 |- ( Base ` S ) = ( Base ` S )
3		eqid 2207	. . . . 5 |- ( Base ` T ) = ( Base ` T )
4	2, 3	ghmf 13698	. . . 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 5460	. 2 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( `' F " V ) C_ ( Base ` S ) )
7		ghmgrp1 13696	. . . . . 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 2207	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
10	2, 9	grpidcl 13476	. . . . 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 2207	. . . . . . 7 |- ( 0g ` T ) = ( 0g ` T )
13	9, 12	ghmid 13700	. . . . . 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 13633	. . . . . 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 2284	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( F ` ( 0g ` S ) ) e. V )
18	5	ffnd 5446	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> F Fn ( Base ` S ) )
19		elpreima 5722	. . . . 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 947	. . 3 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( 0g ` S ) e. ( `' F " V ) )
22		elex2 2793	. . 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 5722	. . . . 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 5722	. . . . . . . . . 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 2207	. . . . . . . . . . 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 537	. . . . . . . . . . 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 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 ) ) ) -> b e. ( Base ` S ) )
33	2, 29, 30, 31, 32	grpcld 13461	. . . . . . . . . 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 2207	. . . . . . . . . . . . 13 |- ( +g ` T ) = ( +g ` T )
36	2, 29, 35	ghmlin 13699	. . . . . . . . . . . 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 1250	. . . . . . . . . . 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 528	. . . . . . . . . . . 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 538	. . . . . . . . . . . 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 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 ` b ) e. V )
41	35	subgcl 13635	. . . . . . . . . . . 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 1250	. . . . . . . . . . 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 2284	. . . . . . . . . 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 5722	. . . . . . . . . . . 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 947	. . . . . . . . 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 2580	. . . . . 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 529	. . . . . . . 8 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) /\ ( a e. ( Base ` S ) /\ ( F ` a ) e. V ) ) -> a e. ( Base ` S ) )
52		eqid 2207	. . . . . . . . 9 |- ( invg ` S ) = ( invg ` S )
53	2, 52	grpinvcl 13495	. . . . . . . 8 |- ( ( S e. Grp /\ a e. ( Base ` S ) ) -> ( ( invg ` S ) ` a ) e. ( Base ` S ) )
54	8, 51, 53	syl2an2r 595	. . . . . . 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 2207	. . . . . . . . . 10 |- ( invg ` T ) = ( invg ` T )
56	2, 52, 55	ghminv 13701	. . . . . . . . 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 13634	. . . . . . . . 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 2284	. . . . . . 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 5722	. . . . . . . . 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 947	. . . . . 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 2580	. 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 13640	. . 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 1183	1 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( `' F " V ) e. (SubGrp ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   E.wex 1516   e. wcel 2178   A.wral 2486   C_ wss 3174   `'ccnv 4692   "cima 4696   Fn wfn 5285   -->wf 5286   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   0gc0g 13203   Grpcgrp 13447   invgcminusg 13448  SubGrpcsubg 13618   GrpHom cghm 13691

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-subg 13621  df-ghm 13692

This theorem is referenced by:  ghmnsgpreima  13720