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Theorem funoprab 6382
Description: "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.)
Hypothesis
Ref Expression
funoprab.1  |-  E* z ph
Assertion
Ref Expression
funoprab  |-  Fun  { <. <. x ,  y
>. ,  z >.  | 
ph }
Distinct variable group:    x, y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem funoprab
StepHypRef Expression
1 funoprab.1 . . 3  |-  E* z ph
21gen2 1640 . 2  |-  A. x A. y E* z ph
3 funoprabg 6381 . 2  |-  ( A. x A. y E* z ph  ->  Fun  { <. <. x ,  y >. ,  z
>.  |  ph } )
42, 3ax-mp 5 1  |-  Fun  { <. <. x ,  y
>. ,  z >.  | 
ph }
Colors of variables: wff setvar class
Syntax hints:   A.wal 1403   E*wmo 2239   Fun wfun 5562   {coprab 6278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-fun 5570  df-oprab 6281
This theorem is referenced by:  mpt2fun  6384  ovidig  6400  ovigg  6403  oprabex  6771  axaddf  9551  axmulf  9552  funtransport  30356  funray  30465  funline  30467
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