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Theorem fnopabg 6478
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 2697 . . . . . 6 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
21albii 1811 . . . . 5 (∀𝑥∃*𝑦(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦𝜑))
3 funopab 6383 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
4 df-ral 3140 . . . . 5 (∀𝑥𝐴 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦𝜑))
52, 3, 43bitr4ri 305 . . . 4 (∀𝑥𝐴 ∃*𝑦𝜑 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
6 dmopab3 5781 . . . 4 (∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
75, 6anbi12i 626 . . 3 ((∀𝑥𝐴 ∃*𝑦𝜑 ∧ ∀𝑥𝐴𝑦𝜑) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴))
8 r19.26 3167 . . 3 (∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ (∀𝑥𝐴 ∃*𝑦𝜑 ∧ ∀𝑥𝐴𝑦𝜑))
9 df-fn 6351 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴))
107, 8, 93bitr4i 304 . 2 (∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴)
11 df-eu 2647 . . . 4 (∃!𝑦𝜑 ↔ (∃𝑦𝜑 ∧ ∃*𝑦𝜑))
1211biancomi 463 . . 3 (∃!𝑦𝜑 ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
1312ralbii 3162 . 2 (∀𝑥𝐴 ∃!𝑦𝜑 ↔ ∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
14 fnopabg.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
1514fneq1i 6443 . 2 (𝐹 Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴)
1610, 13, 153bitr4i 304 1 (∀𝑥𝐴 ∃!𝑦𝜑𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wex 1771  wcel 2105  ∃*wmo 2613  ∃!weu 2646  wral 3135  {copab 5119  dom cdm 5548  Fun wfun 6342   Fn wfn 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-fun 6350  df-fn 6351
This theorem is referenced by:  fnopab  6479  mptfng  6480  axcontlem2  26678
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