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Theorem ghmgrp2 16058
Description: A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Assertion
Ref Expression
ghmgrp2  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )

Proof of Theorem ghmgrp2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2460 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2460 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
3 eqid 2460 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2460 . . . 4  |-  ( +g  `  T )  =  ( +g  `  T )
51, 2, 3, 4isghm 16055 . . 3  |-  ( F  e.  ( S  GrpHom  T )  <->  ( ( S  e.  Grp  /\  T  e.  Grp )  /\  ( F : ( Base `  S
) --> ( Base `  T
)  /\  A. y  e.  ( Base `  S
) A. x  e.  ( Base `  S
) ( F `  ( y ( +g  `  S ) x ) )  =  ( ( F `  y ) ( +g  `  T
) ( F `  x ) ) ) ) )
65simplbi 460 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ( S  e.  Grp  /\  T  e. 
Grp ) )
76simprd 463 1  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   -->wf 5575   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   Grpcgrp 15716    GrpHom cghm 16052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-ghm 16053
This theorem is referenced by:  ghmid  16061  ghminv  16062  ghmmhm  16065  ghmmulg  16067  ghmrn  16068  resghm  16071  ghmco  16074  ghmker  16080  ghmeqker  16081  ghmf1  16083  ghmf1o  16084  ghmpropd  16092  isgim  16098  gicrcl  16109  lactghmga  16217  ghmplusg  16638  ghmcyg  16682  ghmcnp  20341  abliso  27334  gicabl  30640
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