Theorem resghm2b 13598

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 13581	. . 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 13581	. . 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 13532	. . . . . 6 |- ( X e. (SubGrp ` T ) -> X e. (SubMnd ` T ) )
6		resghm2.u	. . . . . . 7 |- U = ( T ↾s X )
7	6	resmhm2b 13321	. . . . . 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 13515	. . . . . . 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 13590	. . . . . 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 2275	. . . 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 13513	. . . . . . 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 13590	. . . . . 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 2275	. . . 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 707	1 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   C_ wss 3166   ran crn 4676   ` cfv 5271  (class class class)co 5944   ↾_s cress 12833   MndHom cmhm 13289  SubMndcsubmnd 13290   Grpcgrp 13332  SubGrpcsubg 13503   GrpHom cghm 13576

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-map 6737  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-mhm 13291  df-submnd 13292  df-grp 13335  df-minusg 13336  df-subg 13506  df-ghm 13577

This theorem is referenced by:  ghmghmrn  13599  resrhm2b  14011