ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funoprab Unicode version

Theorem funoprab 5922
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 1430 . 2  |-  A. x A. y E* z ph
3 funoprabg 5921 . 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 set class
Syntax hints:   A.wal 1333   E*wmo 2007   Fun wfun 5165   {coprab 5826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-fun 5173  df-oprab 5829
This theorem is referenced by:  mpofun  5924  ovidig  5939  ovigg  5942  oprabex  6077  th3qcor  6585  axaddf  7789  axmulf  7790
  Copyright terms: Public domain W3C validator