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

Theorem funopabeq 5159
 Description: A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
Assertion
Ref Expression
funopabeq Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem funopabeq
StepHypRef Expression
1 funopab 5158 . 2 (Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} ↔ ∀𝑥∃*𝑦 𝑦 = 𝐴)
2 moeq 2859 . 2 ∃*𝑦 𝑦 = 𝐴
31, 2mpgbir 1429 1 Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
 Colors of variables: wff set class Syntax hints:   = wceq 1331  ∃*wmo 2000  {copab 3988  Fun wfun 5117 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-fun 5125 This theorem is referenced by:  funopab4  5160
 Copyright terms: Public domain W3C validator