Theorem funoprab 7491

Description: "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.)

Hypothesis

Ref	Expression
funoprab.1	⊢ ∃*𝑧𝜑

Assertion

Ref	Expression
funoprab	⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Distinct variable group:   𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem funoprab

Step	Hyp	Ref	Expression
1		funoprab.1	. . 3 ⊢ ∃*𝑧𝜑
2	1	gen2 1796	. 2 ⊢ ∀𝑥∀𝑦∃*𝑧𝜑
3		funoprabg 7490	. 2 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
4	2, 3	ax-mp 5	1 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:  ∀wal 1538  ∃*wmo 2531  Fun wfun 6493  {coprab 7370

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 5246  ax-nul 5256  ax-pr 5382

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-fun 6501  df-oprab 7373

This theorem is referenced by:  mpofun  7493  ovidig  7511  ovigg  7514  oprabex  7934  axaddf  11074  axmulf  11075  funtransport  35992  funray  36101  funline  36103