MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funopabeq Structured version   Visualization version   GIF version

Theorem funopabeq 5892
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 5891 . 2 (Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} ↔ ∀𝑥∃*𝑦 𝑦 = 𝐴)
2 moeq 3369 . 2 ∃*𝑦 𝑦 = 𝐴
31, 2mpgbir 1723 1 Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  ∃*wmo 2470  {copab 4682  Fun wfun 5851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-fun 5859
This theorem is referenced by:  funopab4  5893
  Copyright terms: Public domain W3C validator