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Theorem fnoprab 6105
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 1553 . 2  |-  A. x A. y ( ph  ->  E! z ps )
3 fnoprabg 6103 . 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 359   A.wal 1546   E!weu 2231   {copab 4199    Fn wfn 5382   {coprab 6014
This theorem is referenced by:  ovid  6122  ov  6125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-fun 5389  df-fn 5390  df-oprab 6017
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