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Theorem ghomgrp 24012
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 4916 . . . . 5  |-  ( ( Y  X.  Y )  =  ( ran  F  X.  ran  F )  <->  Y  =  ran  F )
4 reseq2 4966 . . . . 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 2316 . 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 2296 . . 3  |-  { <. <.
x ,  x >. ,  x >. }  =  { <. <. x ,  x >. ,  x >. }
10 eqid 2296 . . 3  |-  (  _I  |`  { x } )  =  (  _I  |`  { x } )
118, 9, 10ghomgrplem 24011 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  |`  ( ran  F  X.  ran  F ) )  e.  (
SubGrpOp `  H ) )
127, 11syl5eqel 2380 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 1632    e. wcel 1696   {csn 3653   <.cop 3656    _I cid 4320    X. cxp 4703   ran crn 4706    |` cres 4707   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869   SubGrpOpcsubgo 20984   GrpOpHom cghom 21040
This theorem is referenced by:  ghomfo  24013  ghomgsg  24015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-subgo 20985  df-ghom 21041
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