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Theorem fnoprabg 7531
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 2573 . . . . . 6 (∃!𝑧𝜓 → ∃*𝑧𝜓)
21imim2i 16 . . . . 5 ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃*𝑧𝜓))
3 moanimv 2616 . . . . 5 (∃*𝑧(𝜑𝜓) ↔ (𝜑 → ∃*𝑧𝜓))
42, 3sylibr 233 . . . 4 ((𝜑 → ∃!𝑧𝜓) → ∃*𝑧(𝜑𝜓))
542alimi 1815 . . 3 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → ∀𝑥𝑦∃*𝑧(𝜑𝜓))
6 funoprabg 7529 . . 3 (∀𝑥𝑦∃*𝑧(𝜑𝜓) → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)})
75, 6syl 17 . 2 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)})
8 dmoprab 7510 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝜑𝜓)}
9 nfa1 2149 . . . 4 𝑥𝑥𝑦(𝜑 → ∃!𝑧𝜓)
10 nfa2 2171 . . . 4 𝑦𝑥𝑦(𝜑 → ∃!𝑧𝜓)
11 simpl 484 . . . . . . . 8 ((𝜑𝜓) → 𝜑)
1211exlimiv 1934 . . . . . . 7 (∃𝑧(𝜑𝜓) → 𝜑)
13 euex 2572 . . . . . . . . . 10 (∃!𝑧𝜓 → ∃𝑧𝜓)
1413imim2i 16 . . . . . . . . 9 ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃𝑧𝜓))
1514ancld 552 . . . . . . . 8 ((𝜑 → ∃!𝑧𝜓) → (𝜑 → (𝜑 ∧ ∃𝑧𝜓)))
16 19.42v 1958 . . . . . . . 8 (∃𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑧𝜓))
1715, 16imbitrrdi 251 . . . . . . 7 ((𝜑 → ∃!𝑧𝜓) → (𝜑 → ∃𝑧(𝜑𝜓)))
1812, 17impbid2 225 . . . . . 6 ((𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑𝜓) ↔ 𝜑))
1918sps 2179 . . . . 5 (∀𝑦(𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑𝜓) ↔ 𝜑))
2019sps 2179 . . . 4 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → (∃𝑧(𝜑𝜓) ↔ 𝜑))
219, 10, 20opabbid 5214 . . 3 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝜑𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑})
228, 21eqtrid 2785 . 2 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} = {⟨𝑥, 𝑦⟩ ∣ 𝜑})
23 df-fn 6547 . 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 584 1 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wex 1782  ∃*wmo 2533  ∃!weu 2563  {copab 5211  dom cdm 5677  Fun wfun 6538   Fn wfn 6539  {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-dm 5687  df-fun 6546  df-fn 6547  df-oprab 7413
This theorem is referenced by:  fnoprab  7534  ovg  7572
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