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Theorem funfvop 5843
Description: Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
Assertion
Ref Expression
funfvop  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )

Proof of Theorem funfvop
StepHypRef Expression
1 eqid 2437 . 2  |-  ( F `
 A )  =  ( F `  A
)
2 funopfvb 5771 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  =  ( F `  A )  <->  <. A ,  ( F `
 A ) >.  e.  F ) )
31, 2mpbii 204 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   <.cop 3818   dom cdm 4879   Fun wfun 5449   ` cfv 5455
This theorem is referenced by:  funfvbrb  5844  fvimacnv  5846  fnopfv  5866  fvelrn  5867  dff3  5883  funfvima3  5976  tfrlem9a  6648  fundmen  7181  adj1  23437  bnj145  29095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463
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