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Theorem ghomgrp 25102
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 5092 . . . . 5  |-  ( ( Y  X.  Y )  =  ( ran  F  X.  ran  F )  <->  Y  =  ran  F )
4 reseq2 5142 . . . . 5  |-  ( ( Y  X.  Y )  =  ( ran  F  X.  ran  F )  -> 
( H  |`  ( Y  X.  Y ) )  =  ( H  |`  ( ran  F  X.  ran  F ) ) )
53, 4sylbir 206 . . . 4  |-  ( Y  =  ran  F  -> 
( H  |`  ( Y  X.  Y ) )  =  ( H  |`  ( ran  F  X.  ran  F ) ) )
62, 5ax-mp 8 . . 3  |-  ( H  |`  ( Y  X.  Y
) )  =  ( H  |`  ( ran  F  X.  ran  F ) )
71, 6eqtri 2457 . 2  |-  S  =  ( H  |`  ( ran  F  X.  ran  F
) )
8 id 21 . . 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 2437 . . 3  |-  { <. <.
x ,  x >. ,  x >. }  =  { <. <. x ,  x >. ,  x >. }
10 eqid 2437 . . 3  |-  (  _I  |`  { x } )  =  (  _I  |`  { x } )
118, 9, 10ghomgrplem 25101 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  |`  ( ran  F  X.  ran  F ) )  e.  (
SubGrpOp `  H ) )
127, 11syl5eqel 2521 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726   {csn 3815   <.cop 3818    _I cid 4494    X. cxp 4877   ran crn 4880    |` cres 4881   ` cfv 5455  (class class class)co 6082   GrpOpcgr 21775   SubGrpOpcsubgo 21890   GrpOpHom cghom 21946
This theorem is referenced by:  ghomfo  25103  ghomgsg  25105
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-grpo 21780  df-gid 21781  df-ginv 21782  df-subgo 21891  df-ghom 21947
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