Theorem resghm2b 17678

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 17662	. . 3 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2	1	a1i 11	. 2 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) -> S e. Grp ) )
3		ghmgrp1 17662	. . 3 |- ( F e. ( S GrpHom U ) -> S e. Grp )
4	3	a1i 11	. 2 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom U ) -> S e. Grp ) )
5		subgsubm 17616	. . . . . 6 |- ( X e. (SubGrp ` T ) -> X e. (SubMnd ` T ) )
6		resghm2.u	. . . . . . 7 |- U = ( T ↾s X )
7	6	resmhm2b 17361	. . . . . 6 |- ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) -> ( F e. ( S MndHom T ) <-> F e. ( S MndHom U ) ) )
8	5, 7	sylan 488	. . . . 5 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S MndHom T ) <-> F e. ( S MndHom U ) ) )
9	8	adantl 482	. . . 4 |- ( ( S e. Grp /\ ( X e. (SubGrp ` T ) /\ ran F C_ X ) ) -> ( F e. ( S MndHom T ) <-> F e. ( S MndHom U ) ) )
10		subgrcl 17599	. . . . . . 7 |- ( X e. (SubGrp ` T ) -> T e. Grp )
11	10	adantr 481	. . . . . 6 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> T e. Grp )
12		ghmmhmb 17671	. . . . . 6 |- ( ( S e. Grp /\ T e. Grp ) -> ( S GrpHom T ) = ( S MndHom T ) )
13	11, 12	sylan2 491	. . . . 5 |- ( ( S e. Grp /\ ( X e. (SubGrp ` T ) /\ ran F C_ X ) ) -> ( S GrpHom T ) = ( S MndHom T ) )
14	13	eleq2d 2687	. . . 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 17597	. . . . . . 7 |- ( X e. (SubGrp ` T ) -> U e. Grp )
16	15	adantr 481	. . . . . 6 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> U e. Grp )
17		ghmmhmb 17671	. . . . . 6 |- ( ( S e. Grp /\ U e. Grp ) -> ( S GrpHom U ) = ( S MndHom U ) )
18	16, 17	sylan2 491	. . . . 5 |- ( ( S e. Grp /\ ( X e. (SubGrp ` T ) /\ ran F C_ X ) ) -> ( S GrpHom U ) = ( S MndHom U ) )
19	18	eleq2d 2687	. . . 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 300	. . 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 451	. 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 369	1 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ran crn 5115   ` cfv 5888  (class class class)co 6650   ↾_s cress 15858   MndHom cmhm 17333  SubMndcsubmnd 17334   Grpcgrp 17422  SubGrpcsubg 17588   GrpHom cghm 17657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-ghm 17658

This theorem is referenced by:  ghmghmrn  17679  cayley  17834  pj1ghm2  18117  dpjghm2  18463  reslmhm2b  19054  m2cpmghm  20549