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Theorem funopabeq 5479
Description: A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
Assertion
Ref Expression
funopabeq  |-  Fun  { <. x ,  y >.  |  y  =  A }
Distinct variable groups:    x, y    y, A
Allowed substitution hint:    A( x)

Proof of Theorem funopabeq
StepHypRef Expression
1 funopab 5478 . 2  |-  ( Fun 
{ <. x ,  y
>.  |  y  =  A }  <->  A. x E* y 
y  =  A )
2 moeq 3102 . 2  |-  E* y 
y  =  A
31, 2mpgbir 1559 1  |-  Fun  { <. x ,  y >.  |  y  =  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1652   E*wmo 2281   {copab 4257   Fun wfun 5440
This theorem is referenced by:  funopab4  5480
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-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-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-fun 5448
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