Theorem funoprabg 7572

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	⊢ (∀𝑥∀𝑦∃*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.

Step	Hyp	Ref	Expression
1		mosubopt 5489	. . 3 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → ∃*𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2	1	alrimiv 1962	. 2 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → ∀𝑤∃*𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3		dfoprab2 7509	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4	3	funeqi 6588	. . 3 ⊢ (Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ Fun {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)})
5		funopab 6603	. . 3 ⊢ (Fun {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↔ ∀𝑤∃*𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6	4, 5	bitr2i 279	. 2 ⊢ (∀𝑤∃*𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
7	2, 6	sylib 221	1 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812  ∃*wmo 2565  ⟨cop 4840  {copab 5173  Fun wfun 6560  {coprab 7452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-fun 6568  df-oprab 7455

This theorem is used by:  funoprab  7573  fnoprabg  7574  oprabexd  8021