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Theorem ghomcl 23999
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 23998 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X -onto-> Z )
6 fof 5451 . 2  |-  ( F : X -onto-> Z  ->  F : X --> Z )
7 ffvelrn 5663 . . 3  |-  ( ( F : X --> Z  /\  A  e.  X )  ->  ( F `  A
)  e.  Z )
87ex 423 . 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 set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    X. cxp 4687   ran crn 4690    |` cres 4691   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   GrpOpHom cghom 21024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-subgo 20969  df-ghom 21025
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