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Theorem funopab 5881
Description: A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.)
Assertion
Ref Expression
funopab (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem funopab
StepHypRef Expression
1 relopab 5207 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 nfopab1 4681 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
3 nfopab2 4682 . . . 4 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
42, 3dffun6f 5861 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ∀𝑥∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦))
51, 4mpbiran 952 . 2 (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦)
6 df-br 4614 . . . . 5 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
7 opabid 4942 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
86, 7bitri 264 . . . 4 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝜑)
98mobii 2492 . . 3 (∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃*𝑦𝜑)
109albii 1744 . 2 (∀𝑥∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∀𝑥∃*𝑦𝜑)
115, 10bitri 264 1 (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1478  wcel 1987  ∃*wmo 2470  cop 4154   class class class wbr 4613  {copab 4672  Rel wrel 5079  Fun wfun 5841
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 4741  ax-nul 4749  ax-pr 4867
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 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-fun 5849
This theorem is referenced by:  funopabeq  5882  funco  5886  isarep2  5936  mptfnf  5972  fnopabg  5974  opabiotafun  6216  fvopab3ig  6235  opabex  6437  funoprabg  6712  zfrep6  7081  tz7.44lem1  7446  ajfuni  27561  funadj  28591  abrexdomjm  29189  abrexdom  33154
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