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Theorem fnopabg 6706
Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
Hypothesis
Ref Expression
fnopabg.1 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
Assertion
Ref Expression
fnopabg (∀𝑥𝐴 ∃!𝑦𝜑𝐹 Fn 𝐴)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fnopabg
StepHypRef Expression
1 moanimv 2617 . . . . . 6 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
21albii 1816 . . . . 5 (∀𝑥∃*𝑦(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦𝜑))
3 funopab 6603 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
4 df-ral 3060 . . . . 5 (∀𝑥𝐴 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦𝜑))
52, 3, 43bitr4ri 304 . . . 4 (∀𝑥𝐴 ∃*𝑦𝜑 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
6 dmopab3 5933 . . . 4 (∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
75, 6anbi12i 628 . . 3 ((∀𝑥𝐴 ∃*𝑦𝜑 ∧ ∀𝑥𝐴𝑦𝜑) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴))
8 r19.26 3109 . . 3 (∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ (∀𝑥𝐴 ∃*𝑦𝜑 ∧ ∀𝑥𝐴𝑦𝜑))
9 df-fn 6566 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴))
107, 8, 93bitr4i 303 . 2 (∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴)
11 df-eu 2567 . . . 4 (∃!𝑦𝜑 ↔ (∃𝑦𝜑 ∧ ∃*𝑦𝜑))
1211biancomi 462 . . 3 (∃!𝑦𝜑 ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
1312ralbii 3091 . 2 (∀𝑥𝐴 ∃!𝑦𝜑 ↔ ∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
14 fnopabg.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
1514fneq1i 6666 . 2 (𝐹 Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴)
1610, 13, 153bitr4i 303 1 (∀𝑥𝐴 ∃!𝑦𝜑𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  ∃*wmo 2536  ∃!weu 2566  wral 3059  {copab 5210  dom cdm 5689  Fun wfun 6557   Fn wfn 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-fun 6565  df-fn 6566
This theorem is referenced by:  fnopab  6707  mptfng  6708  axcontlem2  28995  tfsconcatfn  43328  tfsconcatfv1  43329  tfsconcatfv2  43330
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