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Theorem fnopabg 5073
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 2018 . . . . . 6 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
21albii 1400 . . . . 5 (∀𝑥∃*𝑦(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦𝜑))
3 funopab 4985 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
4 df-ral 2358 . . . . 5 (∀𝑥𝐴 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦𝜑))
52, 3, 43bitr4ri 211 . . . 4 (∀𝑥𝐴 ∃*𝑦𝜑 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
6 dmopab3 4596 . . . 4 (∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
75, 6anbi12i 448 . . 3 ((∀𝑥𝐴 ∃*𝑦𝜑 ∧ ∀𝑥𝐴𝑦𝜑) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴))
8 r19.26 2490 . . 3 (∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ (∀𝑥𝐴 ∃*𝑦𝜑 ∧ ∀𝑥𝐴𝑦𝜑))
9 df-fn 4955 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴))
107, 8, 93bitr4i 210 . 2 (∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴)
11 eu5 1990 . . . 4 (∃!𝑦𝜑 ↔ (∃𝑦𝜑 ∧ ∃*𝑦𝜑))
12 ancom 262 . . . 4 ((∃𝑦𝜑 ∧ ∃*𝑦𝜑) ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
1311, 12bitri 182 . . 3 (∃!𝑦𝜑 ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
1413ralbii 2377 . 2 (∀𝑥𝐴 ∃!𝑦𝜑 ↔ ∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
15 fnopabg.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
1615fneq1i 5044 . 2 (𝐹 Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴)
1710, 14, 163bitr4i 210 1 (∀𝑥𝐴 ∃!𝑦𝜑𝐹 Fn 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wex 1422  wcel 1434  ∃!weu 1943  ∃*wmo 1944  wral 2353  {copab 3858  dom cdm 4391  Fun wfun 4946   Fn wfn 4947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-fun 4954  df-fn 4955
This theorem is referenced by:  fnopab  5074  mptfng  5075
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