Theorem resghm2b 14065

Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)

Hypothesis

Ref	Expression
resghm2.u	|- U = ( T ↾s X )

Assertion

Ref	Expression
resghm2b	|- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) )

Proof of Theorem resghm2b

Step	Hyp	Ref	Expression
1		ghmgrp1 14048	. . 3 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2	1	a1i 9	. 2 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) -> S e. Grp ) )
3		ghmgrp1 14048	. . 3 |- ( F e. ( S GrpHom U ) -> S e. Grp )
4	3	a1i 9	. 2 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom U ) -> S e. Grp ) )
5		subgsubm 13999	. . . . . 6 |- ( X e. (SubGrp ` T ) -> X e. (SubMnd ` T ) )
6		resghm2.u	. . . . . . 7 |- U = ( T ↾s X )
7	6	resmhm2b 13796	. . . . . 6 |- ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) -> ( F e. ( S MndHom T ) <-> F e. ( S MndHom U ) ) )
8	5, 7	sylan 283	. . . . 5 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S MndHom T ) <-> F e. ( S MndHom U ) ) )
9	8	adantl 277	. . . 4 |- ( ( S e. Grp /\ ( X e. (SubGrp ` T ) /\ ran F C_ X ) ) -> ( F e. ( S MndHom T ) <-> F e. ( S MndHom U ) ) )
10		subgrcl 13982	. . . . . . 7 |- ( X e. (SubGrp ` T ) -> T e. Grp )
11	10	adantr 276	. . . . . 6 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> T e. Grp )
12		ghmmhmb 14057	. . . . . 6 |- ( ( S e. Grp /\ T e. Grp ) -> ( S GrpHom T ) = ( S MndHom T ) )
13	11, 12	sylan2 286	. . . . 5 |- ( ( S e. Grp /\ ( X e. (SubGrp ` T ) /\ ran F C_ X ) ) -> ( S GrpHom T ) = ( S MndHom T ) )
14	13	eleq2d 2308	. . . 4 |- ( ( S e. Grp /\ ( X e. (SubGrp ` T ) /\ ran F C_ X ) ) -> ( F e. ( S GrpHom T ) <-> F e. ( S MndHom T ) ) )
15	6	subggrp 13980	. . . . . . 7 |- ( X e. (SubGrp ` T ) -> U e. Grp )
16	15	adantr 276	. . . . . 6 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> U e. Grp )
17		ghmmhmb 14057	. . . . . 6 |- ( ( S e. Grp /\ U e. Grp ) -> ( S GrpHom U ) = ( S MndHom U ) )
18	16, 17	sylan2 286	. . . . 5 |- ( ( S e. Grp /\ ( X e. (SubGrp ` T ) /\ ran F C_ X ) ) -> ( S GrpHom U ) = ( S MndHom U ) )
19	18	eleq2d 2308	. . . 4 |- ( ( S e. Grp /\ ( X e. (SubGrp ` T ) /\ ran F C_ X ) ) -> ( F e. ( S GrpHom U ) <-> F e. ( S MndHom U ) ) )
20	9, 14, 19	3bitr4d 220	. . 3 |- ( ( S e. Grp /\ ( X e. (SubGrp ` T ) /\ ran F C_ X ) ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) )
21	20	expcom 116	. 2 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( S e. Grp -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) ) )
22	2, 4, 21	pm5.21ndd 717	1 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   C_ wss 3220   ran crn 4775   ` cfv 5377  (class class class)co 6085   ↾_s cress 13353   MndHom cmhm 13764  SubMndcsubmnd 13765   Grpcgrp 13805  SubGrpcsubg 13970   GrpHom cghm 14043

This proof depends on 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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-mhm 13766  df-submnd 13767  df-grp 13808  df-minusg 13809  df-subg 13973  df-ghm 14044

This theorem is used by:  ghmghmrn  14066  resrhm2b  14557