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Theorem fnoprabg 6659
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 2487 . . . . . 6 (∃!𝑧𝜓 → ∃*𝑧𝜓)
21imim2i 16 . . . . 5 ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃*𝑧𝜓))
3 moanimv 2519 . . . . 5 (∃*𝑧(𝜑𝜓) ↔ (𝜑 → ∃*𝑧𝜓))
42, 3sylibr 223 . . . 4 ((𝜑 → ∃!𝑧𝜓) → ∃*𝑧(𝜑𝜓))
542alimi 1731 . . 3 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → ∀𝑥𝑦∃*𝑧(𝜑𝜓))
6 funoprabg 6657 . . 3 (∀𝑥𝑦∃*𝑧(𝜑𝜓) → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)})
75, 6syl 17 . 2 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)})
8 dmoprab 6639 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝜑𝜓)}
9 nfa1 2015 . . . 4 𝑥𝑥𝑦(𝜑 → ∃!𝑧𝜓)
10 nfa2 2027 . . . 4 𝑦𝑥𝑦(𝜑 → ∃!𝑧𝜓)
11 simpl 472 . . . . . . . 8 ((𝜑𝜓) → 𝜑)
1211exlimiv 1845 . . . . . . 7 (∃𝑧(𝜑𝜓) → 𝜑)
13 euex 2482 . . . . . . . . . 10 (∃!𝑧𝜓 → ∃𝑧𝜓)
1413imim2i 16 . . . . . . . . 9 ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃𝑧𝜓))
1514ancld 574 . . . . . . . 8 ((𝜑 → ∃!𝑧𝜓) → (𝜑 → (𝜑 ∧ ∃𝑧𝜓)))
16 19.42v 1905 . . . . . . . 8 (∃𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑧𝜓))
1715, 16syl6ibr 241 . . . . . . 7 ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃𝑧(𝜑𝜓)))
1812, 17impbid2 215 . . . . . 6 ((𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑𝜓) ↔ 𝜑))
1918sps 2043 . . . . 5 (∀𝑦(𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑𝜓) ↔ 𝜑))
2019sps 2043 . . . 4 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑𝜓) ↔ 𝜑))
219, 10, 20opabbid 4647 . . 3 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝜑𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑})
228, 21syl5eq 2656 . 2 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑})
23 df-fn 5807 . 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 695 1 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473   = wceq 1475  wex 1695  ∃!weu 2458  ∃*wmo 2459  {copab 4642  dom cdm 5038  Fun wfun 5798   Fn wfn 5799  {coprab 6550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-fun 5806  df-fn 5807  df-oprab 6553
This theorem is referenced by:  fnoprab  6661  ovg  6697
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