Theorem funoprab 7534

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 1825	. 2 ⊢ ∀𝑥∀𝑦∃*𝑧𝜑
3		funoprabg 7533	. 2 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
4	2, 3	ax-mp 5	1 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

This proof depends on syntax axioms:  ∀wal 1567  ∃*wmo 2564  Fun wfun 6530  {coprab 7413

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-fun 6538  df-oprab 7416

This theorem is used by:  mpofun  7536  ovidig  7554  ovigg  7557  oprabex  7971  axaddf  11136  axmulf  11137  funtransport  36531  funray  36640  funline  36642