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Theorem ghomgrpi 30134
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 2420 . 2  |-  ran  G  =  ran  G
5 eqid 2420 . 2  |-  (GId `  G )  =  (GId
`  G )
6 eqid 2420 . 2  |-  ( inv `  G )  =  ( inv `  G )
7 eqid 2420 . 2  |-  ran  H  =  ran  H
8 eqid 2420 . 2  |-  (GId `  H )  =  (GId
`  H )
9 eqid 2420 . 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 30133 1  |-  S  e.  ( SubGrpOp `  H )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1867    X. cxp 4843   ran crn 4846    |` cres 4847   ` cfv 5592  (class class class)co 6296   GrpOpcgr 25800  GIdcgi 25801   invcgn 25802   SubGrpOpcsubgo 25915   GrpOpHom cghomOLD 25971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-grpo 25805  df-gid 25806  df-ginv 25807  df-subgo 25916  df-ghomOLD 25972
This theorem is referenced by:  ghomgrplem  30136
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