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Theorem ghomcl 24884
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 24883 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X -onto-> Z )
6 fof 5595 . 2  |-  ( F : X -onto-> Z  ->  F : X --> Z )
7 ffvelrn 5809 . . 3  |-  ( ( F : X --> Z  /\  A  e.  X )  ->  ( F `  A
)  e.  Z )
87ex 424 . 2  |-  ( F : X --> Z  -> 
( A  e.  X  ->  ( F `  A
)  e.  Z ) )
95, 6, 83syl 19 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 936    = wceq 1649    e. wcel 1717    X. cxp 4818   ran crn 4821    |` cres 4822   -->wf 5392   -onto->wfo 5394   ` cfv 5396  (class class class)co 6022   GrpOpcgr 21624   GrpOpHom cghom 21795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-grpo 21629  df-gid 21630  df-ginv 21631  df-subgo 21740  df-ghom 21796
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