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Theorem fnpr2ob 17605
Description: Biconditional version of fnpr2o 17604. (Contributed by Jim Kingdon, 27-Sep-2023.)
Assertion
Ref Expression
fnpr2ob ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)

Proof of Theorem fnpr2ob
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fnpr2o 17604 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
2 0ex 5313 . . . . . . . 8 ∅ ∈ V
32prid1 4767 . . . . . . 7 ∅ ∈ {∅, 1o}
4 df2o3 8513 . . . . . . 7 2o = {∅, 1o}
53, 4eleqtrri 2838 . . . . . 6 ∅ ∈ 2o
6 fndm 6672 . . . . . 6 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} = 2o)
75, 6eleqtrrid 2846 . . . . 5 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∅ ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
82eldm2 5915 . . . . 5 (∅ ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ↔ ∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
97, 8sylib 218 . . . 4 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
10 1n0 8525 . . . . . . . . . . 11 1o ≠ ∅
1110nesymi 2996 . . . . . . . . . 10 ¬ ∅ = 1o
12 vex 3482 . . . . . . . . . . 11 𝑘 ∈ V
132, 12opth1 5486 . . . . . . . . . 10 (⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩ → ∅ = 1o)
1411, 13mto 197 . . . . . . . . 9 ¬ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵
15 elpri 4654 . . . . . . . . 9 (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩))
16 orel2 890 . . . . . . . . 9 (¬ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩ → ((⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩) → ⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩))
1714, 15, 16mpsyl 68 . . . . . . . 8 (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩)
182, 12opth 5487 . . . . . . . 8 (⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ↔ (∅ = ∅ ∧ 𝑘 = 𝐴))
1917, 18sylib 218 . . . . . . 7 (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (∅ = ∅ ∧ 𝑘 = 𝐴))
2019simprd 495 . . . . . 6 (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝑘 = 𝐴)
2120eximi 1832 . . . . 5 (∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐴)
22 isset 3492 . . . . 5 (𝐴 ∈ V ↔ ∃𝑘 𝑘 = 𝐴)
2321, 22sylibr 234 . . . 4 (∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝐴 ∈ V)
249, 23syl 17 . . 3 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o𝐴 ∈ V)
25 1oex 8515 . . . . . . . 8 1o ∈ V
2625prid2 4768 . . . . . . 7 1o ∈ {∅, 1o}
2726, 4eleqtrri 2838 . . . . . 6 1o ∈ 2o
2827, 6eleqtrrid 2846 . . . . 5 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → 1o ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
2925eldm2 5915 . . . . 5 (1o ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ↔ ∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
3028, 29sylib 218 . . . 4 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
3110neii 2940 . . . . . . . . . 10 ¬ 1o = ∅
3225, 12opth1 5486 . . . . . . . . . 10 (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ → 1o = ∅)
3331, 32mto 197 . . . . . . . . 9 ¬ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴
34 elpri 4654 . . . . . . . . . 10 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩))
3534orcomd 871 . . . . . . . . 9 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩ ∨ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩))
36 orel2 890 . . . . . . . . 9 (¬ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ → ((⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩ ∨ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩) → ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩))
3733, 35, 36mpsyl 68 . . . . . . . 8 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩)
3825, 12opth 5487 . . . . . . . 8 (⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩ ↔ (1o = 1o𝑘 = 𝐵))
3937, 38sylib 218 . . . . . . 7 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (1o = 1o𝑘 = 𝐵))
4039simprd 495 . . . . . 6 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝑘 = 𝐵)
4140eximi 1832 . . . . 5 (∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐵)
42 isset 3492 . . . . 5 (𝐵 ∈ V ↔ ∃𝑘 𝑘 = 𝐵)
4341, 42sylibr 234 . . . 4 (∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝐵 ∈ V)
4430, 43syl 17 . . 3 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o𝐵 ∈ V)
4524, 44jca 511 . 2 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))
461, 45impbii 209 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 847   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  c0 4339  {cpr 4633  cop 4637  dom cdm 5689   Fn wfn 6558  1oc1o 8498  2oc2o 8499
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-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-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-fun 6565  df-fn 6566  df-om 7888  df-1o 8505  df-2o 8506
This theorem is referenced by:  xpsfrnel2  17611
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