Users' Mathboxes Mathbox for Paul Chapman < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ghomcl Structured version   Unicode version

Theorem ghomcl 28507
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 28506 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X -onto-> Z )
6 fof 5793 . 2  |-  ( F : X -onto-> Z  ->  F : X --> Z )
7 ffvelrn 6017 . . 3  |-  ( ( F : X --> Z  /\  A  e.  X )  ->  ( F `  A
)  e.  Z )
87ex 434 . 2  |-  ( F : X --> Z  -> 
( A  e.  X  ->  ( F `  A
)  e.  Z ) )
95, 6, 83syl 20 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 973    = wceq 1379    e. wcel 1767    X. cxp 4997   ran crn 5000    |` cres 5001   -->wf 5582   -onto->wfo 5584   ` cfv 5586  (class class class)co 6282   GrpOpcgr 24864   GrpOpHom cghom 25035
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 6574
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-grpo 24869  df-gid 24870  df-ginv 24871  df-subgo 24980  df-ghom 25036
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator