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Theorem fnoprab 5740
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 1384 . 2  |-  A. x A. y ( ph  ->  E! z ps )
3 fnoprabg 5738 . 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 1287   E!weu 1948   {copab 3896    Fn wfn 5005   {coprab 5645
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-br 3844  df-opab 3898  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-fun 5012  df-fn 5013  df-oprab 5648
This theorem is referenced by:  ovid  5753  ov  5756
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