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Theorem grpocl 21788
Description: Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1  |-  X  =  ran  G
Assertion
Ref Expression
grpocl  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )

Proof of Theorem grpocl
StepHypRef Expression
1 grpfo.1 . . . 4  |-  X  =  ran  G
21grpofo 21787 . . 3  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
3 fof 5653 . . 3  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
42, 3syl 16 . 2  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) --> X )
5 fovrn 6216 . 2  |-  ( ( G : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A G B )  e.  X
)
64, 5syl3an1 1217 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    X. cxp 4876   ran crn 4879   -->wf 5450   -onto->wfo 5452  (class class class)co 6081   GrpOpcgr 21774
This theorem is referenced by:  grpoidinvlem2  21793  grpoidinvlem3  21794  grpo2grp  21822  grpoinvop  21829  grpodivf  21834  grpomuldivass  21837  grpopnpcan2  21841  gxcl  21853  gxcom  21857  ablo4  21875  gxdi  21884  ghgrp  21956  ghsubgolem  21958  rngogcl  21979  vcgcl  22038  nvgcl  22099  ghomgrpilem2  25097  ghomsn  25099  ghomf1olem  25105  ablo4pnp  26555  ghomco  26558  divrngcl  26573  iscringd  26609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-ov 6084  df-grpo 21779
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