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Theorem funopab 6584
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 relopabv 5822 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 nfopab1 5219 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
3 nfopab2 5220 . . . 4 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
42, 3dffun6f 6562 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ∀𝑥∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦))
51, 4mpbiran 708 . 2 (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦)
6 df-br 5150 . . . . 5 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
7 opabidw 5525 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
86, 7bitri 275 . . . 4 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝜑)
98mobii 2543 . . 3 (∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃*𝑦𝜑)
109albii 1822 . 2 (∀𝑥∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∀𝑥∃*𝑦𝜑)
115, 10bitri 275 1 (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1540  wcel 2107  ∃*wmo 2533  cop 4635   class class class wbr 5149  {copab 5211  Rel wrel 5682  Fun wfun 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-fun 6546
This theorem is referenced by:  funopabeq  6585  funco  6589  isarep2  6640  mptfnf  6686  fnopabg  6688  opabiotafun  6973  fvopab3ig  6995  opabex  7222  funoprabg  7529  zfrep6  7941  tz7.44lem1  8405  pwfir  9176  ajfuni  30112  funadj  31139  abrexdomjm  31744  fineqvrep  34095  satfv0fun  34362  satffunlem1lem1  34393  satffunlem2lem1  34395  abrexdom  36598
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