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Theorem ghomgrpi 28778
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 2467 . 2  |-  ran  G  =  ran  G
5 eqid 2467 . 2  |-  (GId `  G )  =  (GId
`  G )
6 eqid 2467 . 2  |-  ( inv `  G )  =  ( inv `  G )
7 eqid 2467 . 2  |-  ran  H  =  ran  H
8 eqid 2467 . 2  |-  (GId `  H )  =  (GId
`  H )
9 eqid 2467 . 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 28777 1  |-  S  e.  ( SubGrpOp `  H )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767    X. cxp 4997   ran crn 5000    |` cres 5001   ` cfv 5588  (class class class)co 6285   GrpOpcgr 24961  GIdcgi 24962   invcgn 24963   SubGrpOpcsubgo 25076   GrpOpHom cghom 25132
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-grpo 24966  df-gid 24967  df-ginv 24968  df-subgo 25077  df-ghom 25133
This theorem is referenced by:  ghomgrplem  28780
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