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Theorem ghomgrpi 24398
Description: The image of a group homomorphism from  G to  H is a subgroup of  H (inference version). (Contributed by Paul Chapman, 25-Feb-2008.)
Hypotheses
Ref Expression
ghomgrpi.1  |-  G  e. 
GrpOp
ghomgrpi.2  |-  H  e. 
GrpOp
ghomgrpi.3  |-  F  e.  ( G GrpOpHom  H )
ghomgrpi.4  |-  Y  =  ran  F
ghomgrpi.5  |-  S  =  ( H  |`  ( Y  X.  Y ) )
Assertion
Ref Expression
ghomgrpi  |-  S  e.  ( SubGrpOp `  H )

Proof of Theorem ghomgrpi
StepHypRef Expression
1 ghomgrpi.1 . 2  |-  G  e. 
GrpOp
2 ghomgrpi.2 . 2  |-  H  e. 
GrpOp
3 ghomgrpi.3 . 2  |-  F  e.  ( G GrpOpHom  H )
4 eqid 2358 . 2  |-  ran  G  =  ran  G
5 eqid 2358 . 2  |-  (GId `  G )  =  (GId
`  G )
6 eqid 2358 . 2  |-  ( inv `  G )  =  ( inv `  G )
7 eqid 2358 . 2  |-  ran  H  =  ran  H
8 eqid 2358 . 2  |-  (GId `  H )  =  (GId
`  H )
9 eqid 2358 . 2  |-  ( inv `  H )  =  ( inv `  H )
10 ghomgrpi.4 . 2  |-  Y  =  ran  F
11 ghomgrpi.5 . 2  |-  S  =  ( H  |`  ( Y  X.  Y ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11ghomgrpilem2 24397 1  |-  S  e.  ( SubGrpOp `  H )
Colors of variables: wff set class
Syntax hints:    = wceq 1642    e. wcel 1710    X. cxp 4766   ran crn 4769    |` cres 4770   ` cfv 5334  (class class class)co 5942   GrpOpcgr 20959  GIdcgi 20960   invcgn 20961   SubGrpOpcsubgo 21074   GrpOpHom cghom 21130
This theorem is referenced by:  ghomgrplem  24400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-grpo 20964  df-gid 20965  df-ginv 20966  df-subgo 21075  df-ghom 21131
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