Theorem funopab 6393

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

Step	Hyp	Ref	Expression
1		relopab 5699	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		nfopab1 5138	. . . 4 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
3		nfopab2 5139	. . . 4 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
4	2, 3	dffun6f 6372	. . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ∀𝑥∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦))
5	1, 4	mpbiran 707	. 2 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦)
6		df-br 5070	. . . . 5 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
7		opabidw 5415	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
8	6, 7	bitri 277	. . . 4 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑)
9	8	mobii 2630	. . 3 ⊢ (∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃*𝑦𝜑)
10	9	albii 1819	. 2 ⊢ (∀𝑥∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∀𝑥∃*𝑦𝜑)
11	5, 10	bitri 277	1 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑)

Syntax hints:   ↔ wb 208  ∀wal 1534   ∈ wcel 2113  ∃*wmo 2619  ⟨cop 4576   class class class wbr 5069  {copab 5131  Rel wrel 5563  Fun wfun 6352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-fun 6360

This theorem is referenced by:  funopabeq  6394  funco  6398  isarep2  6446  mptfnf  6486  fnopabg  6488  opabiotafun  6747  fvopab3ig  6767  opabex  6986  funoprabg  7276  zfrep6  7659  tz7.44lem1  8044  ajfuni  28639  funadj  29666  abrexdomjm  30270  satfv0fun  32622  satffunlem1lem1  32653  satffunlem2lem1  32655  abrexdom  35009