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Theorem ghomgrp 23997
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 4900 . . . . 5  |-  ( ( Y  X.  Y )  =  ( ran  F  X.  ran  F )  <->  Y  =  ran  F )
4 reseq2 4950 . . . . 5  |-  ( ( Y  X.  Y )  =  ( ran  F  X.  ran  F )  -> 
( H  |`  ( Y  X.  Y ) )  =  ( H  |`  ( ran  F  X.  ran  F ) ) )
53, 4sylbir 204 . . . 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 2303 . 2  |-  S  =  ( H  |`  ( ran  F  X.  ran  F
) )
8 id 19 . . 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 2283 . . 3  |-  { <. <.
x ,  x >. ,  x >. }  =  { <. <. x ,  x >. ,  x >. }
10 eqid 2283 . . 3  |-  (  _I  |`  { x } )  =  (  _I  |`  { x } )
118, 9, 10ghomgrplem 23996 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  |`  ( ran  F  X.  ran  F ) )  e.  (
SubGrpOp `  H ) )
127, 11syl5eqel 2367 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 934    = wceq 1623    e. wcel 1684   {csn 3640   <.cop 3643    _I cid 4304    X. cxp 4687   ran crn 4690    |` cres 4691   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   SubGrpOpcsubgo 20968   GrpOpHom cghom 21024
This theorem is referenced by:  ghomfo  23998  ghomgsg  24000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-subgo 20969  df-ghom 21025
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