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Theorem fnopab 5074
Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
Hypotheses
Ref Expression
fnopab.1  |-  ( x  e.  A  ->  E! y ph )
fnopab.2  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }
Assertion
Ref Expression
fnopab  |-  F  Fn  A
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)    F( x, y)

Proof of Theorem fnopab
StepHypRef Expression
1 fnopab.1 . . 3  |-  ( x  e.  A  ->  E! y ph )
21rgen 2421 . 2  |-  A. x  e.  A  E! y ph
3 fnopab.2 . . 3  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }
43fnopabg 5073 . 2  |-  ( A. x  e.  A  E! y ph  <->  F  Fn  A
)
52, 4mpbi 143 1  |-  F  Fn  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   E!weu 1943   A.wral 2353   {copab 3858    Fn wfn 4947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-fun 4954  df-fn 4955
This theorem is referenced by:  fvopab3g  5298
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