Theorem funoprabg 7529

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 5511	. . 3 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → ∃*𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2	1	alrimiv 1931	. 2 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → ∀𝑤∃*𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3		dfoprab2 7467	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4	3	funeqi 6570	. . 3 ⊢ (Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ Fun {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)})
5		funopab 6584	. . 3 ⊢ (Fun {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↔ ∀𝑤∃*𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6	4, 5	bitr2i 276	. 2 ⊢ (∀𝑤∃*𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
7	2, 6	sylib 217	1 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1540   = wceq 1542  ∃wex 1782  ∃*wmo 2533  ⟨cop 4635  {copab 5211  Fun wfun 6538  {coprab 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-fun 6546  df-oprab 7413

This theorem is referenced by:  funoprab  7530  fnoprabg  7531  oprabexd  7962