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Theorem resghm2b 15026
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  =  ( Ts  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
StepHypRef Expression
1 ghmgrp1 15010 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
21a1i 11 . 2  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
)
3 ghmgrp1 15010 . . 3  |-  ( F  e.  ( S  GrpHom  U )  ->  S  e.  Grp )
43a1i 11 . 2  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S  GrpHom  U )  ->  S  e.  Grp )
)
5 subgsubm 14964 . . . . . 6  |-  ( X  e.  (SubGrp `  T
)  ->  X  e.  (SubMnd `  T ) )
6 resghm2.u . . . . . . 7  |-  U  =  ( Ts  X )
76resmhm2b 14763 . . . . . 6  |-  ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S MndHom  T )  <->  F  e.  ( S MndHom  U ) ) )
85, 7sylan 459 . . . . 5  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S MndHom  T )  <->  F  e.  ( S MndHom  U ) ) )
98adantl 454 . . . 4  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( F  e.  ( S MndHom  T )  <-> 
F  e.  ( S MndHom  U ) ) )
10 subgrcl 14951 . . . . . . 7  |-  ( X  e.  (SubGrp `  T
)  ->  T  e.  Grp )
1110adantr 453 . . . . . 6  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  ->  T  e.  Grp )
12 ghmmhmb 15019 . . . . . 6  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( S  GrpHom  T )  =  ( S MndHom  T
) )
1311, 12sylan2 462 . . . . 5  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( S  GrpHom  T )  =  ( S MndHom  T ) )
1413eleq2d 2505 . . . 4  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( F  e.  ( S  GrpHom  T )  <-> 
F  e.  ( S MndHom  T ) ) )
156subggrp 14949 . . . . . . 7  |-  ( X  e.  (SubGrp `  T
)  ->  U  e.  Grp )
1615adantr 453 . . . . . 6  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  ->  U  e.  Grp )
17 ghmmhmb 15019 . . . . . 6  |-  ( ( S  e.  Grp  /\  U  e.  Grp )  ->  ( S  GrpHom  U )  =  ( S MndHom  U
) )
1816, 17sylan2 462 . . . . 5  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( S  GrpHom  U )  =  ( S MndHom  U ) )
1918eleq2d 2505 . . . 4  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( F  e.  ( S  GrpHom  U )  <-> 
F  e.  ( S MndHom  U ) ) )
209, 14, 193bitr4d 278 . . 3  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( F  e.  ( S  GrpHom  T )  <-> 
F  e.  ( S 
GrpHom  U ) ) )
2120expcom 426 . 2  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( S  e.  Grp  ->  ( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) ) )
222, 4, 21pm5.21ndd 345 1  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   ran crn 4881   ` cfv 5456  (class class class)co 6083   ↾s cress 13472   Grpcgrp 14687   MndHom cmhm 14738  SubMndcsubmnd 14739  SubGrpcsubg 14940    GrpHom cghm 15005
This theorem is referenced by:  cayley  15114  pj1ghm2  15338  dpjghm2  15624  reslmhm2b  16132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mnd 14692  df-mhm 14740  df-submnd 14741  df-grp 14814  df-minusg 14815  df-subg 14943  df-ghm 15006
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