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Theorem ghomgrp 28855
Description: The image of a group homomorphism from  G to  H is a subgroup of  H. (Contributed by Paul Chapman, 25-Feb-2008.)
Hypotheses
Ref Expression
ghomgrp.1  |-  Y  =  ran  F
ghomgrp.2  |-  S  =  ( H  |`  ( Y  X.  Y ) )
Assertion
Ref Expression
ghomgrp  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )

Proof of Theorem ghomgrp
StepHypRef Expression
1 ghomgrp.2 . . 3  |-  S  =  ( H  |`  ( Y  X.  Y ) )
2 ghomgrp.1 . . . 4  |-  Y  =  ran  F
3 xpid11 5230 . . . . 5  |-  ( ( Y  X.  Y )  =  ( ran  F  X.  ran  F )  <->  Y  =  ran  F )
4 reseq2 5274 . . . . 5  |-  ( ( Y  X.  Y )  =  ( ran  F  X.  ran  F )  -> 
( H  |`  ( Y  X.  Y ) )  =  ( H  |`  ( ran  F  X.  ran  F ) ) )
53, 4sylbir 213 . . . 4  |-  ( Y  =  ran  F  -> 
( H  |`  ( Y  X.  Y ) )  =  ( H  |`  ( ran  F  X.  ran  F ) ) )
62, 5ax-mp 5 . . 3  |-  ( H  |`  ( Y  X.  Y
) )  =  ( H  |`  ( ran  F  X.  ran  F ) )
71, 6eqtri 2496 . 2  |-  S  =  ( H  |`  ( ran  F  X.  ran  F
) )
8 id 22 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) ) )
9 eqid 2467 . . 3  |-  { <. <.
x ,  x >. ,  x >. }  =  { <. <. x ,  x >. ,  x >. }
10 eqid 2467 . . 3  |-  (  _I  |`  { x } )  =  (  _I  |`  { x } )
118, 9, 10ghomgrplem 28854 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  |`  ( ran  F  X.  ran  F ) )  e.  (
SubGrpOp `  H ) )
127, 11syl5eqel 2559 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   {csn 4033   <.cop 4039    _I cid 4796    X. cxp 5003   ran crn 5006    |` cres 5007   ` cfv 5594  (class class class)co 6295   GrpOpcgr 25011   SubGrpOpcsubgo 25126   GrpOpHom cghom 25182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-grpo 25016  df-gid 25017  df-ginv 25018  df-subgo 25127  df-ghom 25183
This theorem is referenced by:  ghomfo  28856  ghomgsg  28858
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