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Theorem fnopabg 5241
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 2072 . . . . . 6 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
21albii 1446 . . . . 5 (∀𝑥∃*𝑦(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦𝜑))
3 funopab 5153 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
4 df-ral 2419 . . . . 5 (∀𝑥𝐴 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦𝜑))
52, 3, 43bitr4ri 212 . . . 4 (∀𝑥𝐴 ∃*𝑦𝜑 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
6 dmopab3 4747 . . . 4 (∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
75, 6anbi12i 455 . . 3 ((∀𝑥𝐴 ∃*𝑦𝜑 ∧ ∀𝑥𝐴𝑦𝜑) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴))
8 r19.26 2556 . . 3 (∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ (∀𝑥𝐴 ∃*𝑦𝜑 ∧ ∀𝑥𝐴𝑦𝜑))
9 df-fn 5121 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴))
107, 8, 93bitr4i 211 . 2 (∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴)
11 eu5 2044 . . . 4 (∃!𝑦𝜑 ↔ (∃𝑦𝜑 ∧ ∃*𝑦𝜑))
12 ancom 264 . . . 4 ((∃𝑦𝜑 ∧ ∃*𝑦𝜑) ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
1311, 12bitri 183 . . 3 (∃!𝑦𝜑 ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
1413ralbii 2439 . 2 (∀𝑥𝐴 ∃!𝑦𝜑 ↔ ∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
15 fnopabg.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
1615fneq1i 5212 . 2 (𝐹 Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴)
1710, 14, 163bitr4i 211 1 (∀𝑥𝐴 ∃!𝑦𝜑𝐹 Fn 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1329   = wceq 1331  wex 1468  wcel 1480  ∃!weu 1997  ∃*wmo 1998  wral 2414  {copab 3983  dom cdm 4534  Fun wfun 5112   Fn wfn 5113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-fun 5120  df-fn 5121
This theorem is referenced by:  fnopab  5242  mptfng  5243
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