Theorem funoprab 6658

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 1714	. 2 ⊢ ∀𝑥∀𝑦∃*𝑧𝜑
3		funoprabg 6657	. 2 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
4	2, 3	ax-mp 5	1 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:  ∀wal 1473  ∃*wmo 2459  Fun wfun 5798  {coprab 6550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806  df-oprab 6553

This theorem is referenced by:  mpt2fun  6660  ovidig  6676  ovigg  6679  oprabex  7047  axaddf  9845  axmulf  9846  funtransport  31308  funray  31417  funline  31419