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Theorem ghomcl 29884
Description: Closure of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.)
Hypotheses
Ref Expression
ghomfo.1  |-  X  =  ran  G
ghomfo.2  |-  Y  =  ran  F
ghomfo.3  |-  S  =  ( H  |`  ( Y  X.  Y ) )
ghomfo.4  |-  Z  =  ran  S
Assertion
Ref Expression
ghomcl  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( A  e.  X  ->  ( F `  A )  e.  Z
) )

Proof of Theorem ghomcl
StepHypRef Expression
1 ghomfo.1 . . 3  |-  X  =  ran  G
2 ghomfo.2 . . 3  |-  Y  =  ran  F
3 ghomfo.3 . . 3  |-  S  =  ( H  |`  ( Y  X.  Y ) )
4 ghomfo.4 . . 3  |-  Z  =  ran  S
51, 2, 3, 4ghomfo 29883 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X -onto-> Z )
6 fof 5778 . 2  |-  ( F : X -onto-> Z  ->  F : X --> Z )
7 ffvelrn 6007 . . 3  |-  ( ( F : X --> Z  /\  A  e.  X )  ->  ( F `  A
)  e.  Z )
87ex 432 . 2  |-  ( F : X --> Z  -> 
( A  e.  X  ->  ( F `  A
)  e.  Z ) )
95, 6, 83syl 18 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( A  e.  X  ->  ( F `  A )  e.  Z
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842    X. cxp 4821   ran crn 4824    |` cres 4825   -->wf 5565   -onto->wfo 5567   ` cfv 5569  (class class class)co 6278   GrpOpcgr 25602   GrpOpHom cghomOLD 25773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-grpo 25607  df-gid 25608  df-ginv 25609  df-subgo 25718  df-ghomOLD 25774
This theorem is referenced by: (None)
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