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Theorem fnoprab 5635
Description: Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.)
Hypothesis
Ref Expression
fnoprab.1  |-  ( ph  ->  E! z ps )
Assertion
Ref Expression
fnoprab  |-  { <. <.
x ,  y >. ,  z >.  |  (
ph  /\  ps ) }  Fn  { <. x ,  y >.  |  ph }
Distinct variable groups:    x, y, z    ph, z
Allowed substitution hints:    ph( x, y)    ps( x, y, z)

Proof of Theorem fnoprab
StepHypRef Expression
1 fnoprab.1 . . 3  |-  ( ph  ->  E! z ps )
21gen2 1380 . 2  |-  A. x A. y ( ph  ->  E! z ps )
3 fnoprabg 5633 . 2  |-  ( A. x A. y ( ph  ->  E! z ps )  ->  { <. <. x ,  y
>. ,  z >.  |  ( ph  /\  ps ) }  Fn  { <. x ,  y >.  |  ph } )
42, 3ax-mp 7 1  |-  { <. <.
x ,  y >. ,  z >.  |  (
ph  /\  ps ) }  Fn  { <. x ,  y >.  |  ph }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283   E!weu 1942   {copab 3846    Fn wfn 4927   {coprab 5544
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-fun 4934  df-fn 4935  df-oprab 5547
This theorem is referenced by:  ovid  5648  ov  5651
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