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Theorem ghmgrp2 15009
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 2436 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2436 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
3 eqid 2436 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2436 . . . 4  |-  ( +g  `  T )  =  ( +g  `  T )
51, 2, 3, 4isghm 15006 . . 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 447 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ( S  e.  Grp  /\  T  e. 
Grp ) )
76simprd 450 1  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   Grpcgrp 14685    GrpHom cghm 15003
This theorem is referenced by:  ghmid  15012  ghminv  15013  ghmmhm  15016  ghmmulg  15018  ghmrn  15019  resghm  15022  ghmco  15025  ghmker  15031  ghmeqker  15032  ghmf1  15034  ghmf1o  15035  ghmpropd  15043  isgim  15049  gicrcl  15060  lactghmga  15107  ghmplusg  15461  ghmcyg  15505  ghmcnp  18144  gicabl  27240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-ghm 15004
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