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Theorem funopabeq 5221
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 5220 . 2  |-  ( Fun 
{ <. x ,  y
>.  |  y  =  A }  <->  A. x E* y 
y  =  A )
2 moeq 2899 . 2  |-  E* y 
y  =  A
31, 2mpgbir 1440 1  |-  Fun  { <. x ,  y >.  |  y  =  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1342   E*wmo 2014   {copab 4039   Fun wfun 5179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2726  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-br 3980  df-opab 4041  df-id 4268  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-fun 5187
This theorem is referenced by:  funopab4  5222
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