Theorem funopab 6521

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		relopabv 5768	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		nfopab1 5165	. . . 4 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
3		nfopab2 5166	. . . 4 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
4	2, 3	dffun6f 6501	. . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ∀𝑥∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦))
5	1, 4	mpbiran 709	. 2 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∃*𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦)
6		df-br 5096	. . . . 5 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
7		opabidw 5471	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
8	6, 7	bitri 275	. . . 4 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑)
9	8	mobii 2541	. . 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 275	1 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1538   ∈ wcel 2109  ∃*wmo 2531  ⟨cop 4585   class class class wbr 5095  {copab 5157  Rel wrel 5628  Fun wfun 6480

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-fun 6488

This theorem is referenced by:  funopabeq  6522  funco  6526  isarep2  6576  mptfnf  6621  fnopabg  6623  opabiotafun  6907  fvopab3ig  6930  opabex  7160  funoprabg  7474  zfrep6  7897  tz7.44lem1  8334  pwfir  9224  ajfuni  30821  funadj  31848  abrexdomjm  32469  fineqvrep  35069  satfv0fun  35343  satffunlem1lem1  35374  satffunlem2lem1  35376  abrexdom  37709  modelaxreplem2  44953