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Theorem fnoprab 5947
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 1534 . 2  |-  A. x A. y ( ph  ->  E! z ps )
3 fnoprabg 5945 . 2  |-  ( A. x A. y ( ph  ->  E! z ps )  ->  { <. <. x ,  y
>. ,  z >.  |  ( ph  /\  ps ) }  Fn  { <. x ,  y >.  |  ph } )
42, 3ax-mp 8 1  |-  { <. <.
x ,  y >. ,  z >.  |  (
ph  /\  ps ) }  Fn  { <. x ,  y >.  |  ph }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527   E!weu 2143   {copab 4076    Fn wfn 5250   {coprab 5859
This theorem is referenced by:  ovid  5964  ov  5967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258  df-oprab 5862
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