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Theorem funopabeq 5428
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 5427 . 2  |-  ( Fun 
{ <. x ,  y
>.  |  y  =  A }  <->  A. x E* y 
y  =  A )
2 moeq 3054 . 2  |-  E* y 
y  =  A
31, 2mpgbir 1556 1  |-  Fun  { <. x ,  y >.  |  y  =  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1649   E*wmo 2240   {copab 4207   Fun wfun 5389
This theorem is referenced by:  funopab4  5429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-fun 5397
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