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Theorem funfvop 5598
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 2284 . 2  |-  ( F `
 A )  =  ( F `  A
)
2 funopfvb 5527 . 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 6    /\ wa 360    = wceq 1624    e. wcel 1685   <.cop 3644   dom cdm 4688   Fun wfun 5215   ` cfv 5221
This theorem is referenced by:  funfvbrb  5599  fvimacnv  5601  fnopfv  5621  fvelrn  5622  dff3  5634  funfvima3  5716  tfrlem9a  6397  fundmen  6929  adj1  22505  bnj145  28022
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-fv 5229
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