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Theorem funopfvb 5762
Description: Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)
Assertion
Ref Expression
funopfvb  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  =  B  <->  <. A ,  B >.  e.  F ) )

Proof of Theorem funopfvb
StepHypRef Expression
1 funfn 5474 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fnopfvb 5760 . 2  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  ( ( F `  A )  =  B  <->  <. A ,  B >.  e.  F ) )
31, 2sylanb 459 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  =  B  <->  <. A ,  B >.  e.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3809   dom cdm 4870   Fun wfun 5440    Fn wfn 5441   ` cfv 5446
This theorem is referenced by:  dmfco  5789  funfvop  5834  f1eqcocnv  6020  usgraedgop  21373
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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