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Theorem funopab 6580
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 5819 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 nfopab1 5217 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
3 nfopab2 5218 . . . 4 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
42, 3dffun6f 6558 . . 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 5148 . . . . 5 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
7 opabidw 5523 . . . . 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 4633   class class class wbr 5147  {copab 5209  Rel wrel 5680  Fun wfun 6534
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 5298  ax-nul 5305  ax-pr 5426
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 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-fun 6542
This theorem is referenced by:  funopabeq  6581  funco  6585  isarep2  6636  mptfnf  6682  fnopabg  6684  opabiotafun  6968  fvopab3ig  6990  opabex  7217  funoprabg  7524  zfrep6  7936  tz7.44lem1  8400  pwfir  9172  ajfuni  30090  funadj  31117  abrexdomjm  31722  fineqvrep  34033  satfv0fun  34300  satffunlem1lem1  34331  satffunlem2lem1  34333  abrexdom  36536
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