Theorem funoprab 7514

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 7513	. 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 2532  Fun wfun 6508  {coprab 7391

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-fun 6516  df-oprab 7394

This theorem is referenced by:  mpofun  7516  ovidig  7534  ovigg  7537  oprabex  7958  axaddf  11105  axmulf  11106  funtransport  36026  funray  36135  funline  36137