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Theorem funoprabg 6801
Description: "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.)
Assertion
Ref Expression
funoprabg (∀𝑥𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem funoprabg
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 mosubopt 5001 . . 3 (∀𝑥𝑦∃*𝑧𝜑 → ∃*𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
21alrimiv 1895 . 2 (∀𝑥𝑦∃*𝑧𝜑 → ∀𝑤∃*𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3 dfoprab2 6743 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
43funeqi 5947 . . 3 (Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ Fun {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)})
5 funopab 5961 . . 3 (Fun {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↔ ∀𝑤∃*𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
64, 5bitr2i 265 . 2 (∀𝑤∃*𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
72, 6sylib 208 1 (∀𝑥𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521   = wceq 1523  wex 1744  ∃*wmo 2499  cop 4216  {copab 4745  Fun wfun 5920  {coprab 6691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-fun 5928  df-oprab 6694
This theorem is referenced by:  funoprab  6802  fnoprabg  6803  oprabexd  7197
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