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Theorem funoprab 7267
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
StepHypRef Expression
1 funoprab.1 . . 3 ∃*𝑧𝜑
21gen2 1798 . 2 𝑥𝑦∃*𝑧𝜑
3 funoprabg 7266 . 2 (∀𝑥𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
42, 3ax-mp 5 1 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wal 1536  ∃*wmo 2622  Fun wfun 6337  {coprab 7150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-fun 6345  df-oprab 7153
This theorem is referenced by:  mpofun  7269  ovidig  7285  ovigg  7288  oprabex  7672  axaddf  10565  axmulf  10566  funtransport  33549  funray  33658  funline  33660
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