Theorem funoprab 7572

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 1794	. 2 ⊢ ∀𝑥∀𝑦∃*𝑧𝜑
3		funoprabg 7571	. 2 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
4	2, 3	ax-mp 5	1 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:  ∀wal 1535  ∃*wmo 2541  Fun wfun 6567  {coprab 7449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-fun 6575  df-oprab 7452

This theorem is referenced by:  mpofun  7574  ovidig  7592  ovigg  7595  oprabex  8017  axaddf  11214  axmulf  11215  funtransport  35995  funray  36104  funline  36106