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Theorem fnoprabg 6803
 Description: Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)
Assertion
Ref Expression
fnoprabg (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem fnoprabg
StepHypRef Expression
1 eumo 2527 . . . . . 6 (∃!𝑧𝜓 → ∃*𝑧𝜓)
21imim2i 16 . . . . 5 ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃*𝑧𝜓))
3 moanimv 2560 . . . . 5 (∃*𝑧(𝜑𝜓) ↔ (𝜑 → ∃*𝑧𝜓))
42, 3sylibr 224 . . . 4 ((𝜑 → ∃!𝑧𝜓) → ∃*𝑧(𝜑𝜓))
542alimi 1780 . . 3 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → ∀𝑥𝑦∃*𝑧(𝜑𝜓))
6 funoprabg 6801 . . 3 (∀𝑥𝑦∃*𝑧(𝜑𝜓) → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)})
75, 6syl 17 . 2 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)})
8 dmoprab 6783 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝜑𝜓)}
9 nfa1 2068 . . . 4 𝑥𝑥𝑦(𝜑 → ∃!𝑧𝜓)
10 nfa2 2080 . . . 4 𝑦𝑥𝑦(𝜑 → ∃!𝑧𝜓)
11 simpl 472 . . . . . . . 8 ((𝜑𝜓) → 𝜑)
1211exlimiv 1898 . . . . . . 7 (∃𝑧(𝜑𝜓) → 𝜑)
13 euex 2522 . . . . . . . . . 10 (∃!𝑧𝜓 → ∃𝑧𝜓)
1413imim2i 16 . . . . . . . . 9 ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃𝑧𝜓))
1514ancld 575 . . . . . . . 8 ((𝜑 → ∃!𝑧𝜓) → (𝜑 → (𝜑 ∧ ∃𝑧𝜓)))
16 19.42v 1921 . . . . . . . 8 (∃𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑧𝜓))
1715, 16syl6ibr 242 . . . . . . 7 ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃𝑧(𝜑𝜓)))
1812, 17impbid2 216 . . . . . 6 ((𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑𝜓) ↔ 𝜑))
1918sps 2093 . . . . 5 (∀𝑦(𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑𝜓) ↔ 𝜑))
2019sps 2093 . . . 4 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑𝜓) ↔ 𝜑))
219, 10, 20opabbid 4748 . . 3 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝜑𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑})
228, 21syl5eq 2697 . 2 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑})
23 df-fn 5929 . 2 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ (Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} ∧ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
247, 22, 23sylanbrc 699 1 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521   = wceq 1523  ∃wex 1744  ∃!weu 2498  ∃*wmo 2499  {copab 4745  dom cdm 5143  Fun wfun 5920   Fn wfn 5921  {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-dm 5153  df-fun 5928  df-fn 5929  df-oprab 6694 This theorem is referenced by:  fnoprab  6805  ovg  6841
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