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Theorem fnopabg 5022
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 1975 . . . . . 6 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
21albii 1359 . . . . 5 (∀𝑥∃*𝑦(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦𝜑))
3 funopab 4935 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
4 df-ral 2311 . . . . 5 (∀𝑥𝐴 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦𝜑))
52, 3, 43bitr4ri 202 . . . 4 (∀𝑥𝐴 ∃*𝑦𝜑 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
6 dmopab3 4548 . . . 4 (∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
75, 6anbi12i 433 . . 3 ((∀𝑥𝐴 ∃*𝑦𝜑 ∧ ∀𝑥𝐴𝑦𝜑) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴))
8 r19.26 2441 . . 3 (∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ (∀𝑥𝐴 ∃*𝑦𝜑 ∧ ∀𝑥𝐴𝑦𝜑))
9 df-fn 4905 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴))
107, 8, 93bitr4i 201 . 2 (∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴)
11 eu5 1947 . . . 4 (∃!𝑦𝜑 ↔ (∃𝑦𝜑 ∧ ∃*𝑦𝜑))
12 ancom 253 . . . 4 ((∃𝑦𝜑 ∧ ∃*𝑦𝜑) ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
1311, 12bitri 173 . . 3 (∃!𝑦𝜑 ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
1413ralbii 2330 . 2 (∀𝑥𝐴 ∃!𝑦𝜑 ↔ ∀𝑥𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
15 fnopabg.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
1615fneq1i 4993 . 2 (𝐹 Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} Fn 𝐴)
1710, 14, 163bitr4i 201 1 (∀𝑥𝐴 ∃!𝑦𝜑𝐹 Fn 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wal 1241   = wceq 1243  wex 1381  wcel 1393  ∃!weu 1900  ∃*wmo 1901  wral 2306  {copab 3817  dom cdm 4345  Fun wfun 4896   Fn wfn 4897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-fun 4904  df-fn 4905
This theorem is referenced by:  fnopab  5023  mptfng  5024
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