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Theorem fnoprab 5963
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 1537 . 2  |-  A. x A. y ( ph  ->  E! z ps )
3 fnoprabg 5961 . 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 1530   E!weu 2156   {copab 4092    Fn wfn 5266   {coprab 5875
This theorem is referenced by:  ovid  5980  ov  5983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274  df-oprab 5878
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