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Theorem fnpr2ob 17603
Description: Biconditional version of fnpr2o 17602. (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 17602 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
2 0ex 5307 . . . . . . . 8 ∅ ∈ V
32prid1 4762 . . . . . . 7 ∅ ∈ {∅, 1o}
4 df2o3 8514 . . . . . . 7 2o = {∅, 1o}
53, 4eleqtrri 2840 . . . . . 6 ∅ ∈ 2o
6 fndm 6671 . . . . . 6 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} = 2o)
75, 6eleqtrrid 2848 . . . . 5 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∅ ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
82eldm2 5912 . . . . 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 8526 . . . . . . . . . . 11 1o ≠ ∅
1110nesymi 2998 . . . . . . . . . 10 ¬ ∅ = 1o
12 vex 3484 . . . . . . . . . . 11 𝑘 ∈ V
132, 12opth1 5480 . . . . . . . . . 10 (⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩ → ∅ = 1o)
1411, 13mto 197 . . . . . . . . 9 ¬ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵
15 elpri 4649 . . . . . . . . 9 (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩))
16 orel2 891 . . . . . . . . 9 (¬ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩ → ((⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩) → ⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩))
1714, 15, 16mpsyl 68 . . . . . . . 8 (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩)
182, 12opth 5481 . . . . . . . 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 1835 . . . . 5 (∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐴)
22 isset 3494 . . . . 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 8516 . . . . . . . 8 1o ∈ V
2625prid2 4763 . . . . . . 7 1o ∈ {∅, 1o}
2726, 4eleqtrri 2840 . . . . . 6 1o ∈ 2o
2827, 6eleqtrrid 2848 . . . . 5 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → 1o ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
2925eldm2 5912 . . . . 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 2942 . . . . . . . . . 10 ¬ 1o = ∅
3225, 12opth1 5480 . . . . . . . . . 10 (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ → 1o = ∅)
3331, 32mto 197 . . . . . . . . 9 ¬ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴
34 elpri 4649 . . . . . . . . . 10 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩))
3534orcomd 872 . . . . . . . . 9 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩ ∨ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩))
36 orel2 891 . . . . . . . . 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 5481 . . . . . . . 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 1835 . . . . 5 (∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐵)
42 isset 3494 . . . . 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 848   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  c0 4333  {cpr 4628  cop 4632  dom cdm 5685   Fn wfn 6556  1oc1o 8499  2oc2o 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-fun 6563  df-fn 6564  df-om 7888  df-1o 8506  df-2o 8507
This theorem is referenced by:  xpsfrnel2  17609
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