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Theorem fnpr2ob 17441
Description: Biconditional version of fnpr2o 17440. (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 17440 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
2 0ex 5265 . . . . . . . 8 ∅ ∈ V
32prid1 4724 . . . . . . 7 ∅ ∈ {∅, 1o}
4 df2o3 8421 . . . . . . 7 2o = {∅, 1o}
53, 4eleqtrri 2837 . . . . . 6 ∅ ∈ 2o
6 fndm 6606 . . . . . 6 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} = 2o)
75, 6eleqtrrid 2845 . . . . 5 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∅ ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
82eldm2 5858 . . . . 5 (∅ ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ↔ ∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
97, 8sylib 217 . . . 4 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
10 1n0 8435 . . . . . . . . . . 11 1o ≠ ∅
1110nesymi 3002 . . . . . . . . . 10 ¬ ∅ = 1o
12 vex 3450 . . . . . . . . . . 11 𝑘 ∈ V
132, 12opth1 5433 . . . . . . . . . 10 (⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩ → ∅ = 1o)
1411, 13mto 196 . . . . . . . . 9 ¬ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵
15 elpri 4609 . . . . . . . . 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 5434 . . . . . . . 8 (⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ↔ (∅ = ∅ ∧ 𝑘 = 𝐴))
1917, 18sylib 217 . . . . . . 7 (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (∅ = ∅ ∧ 𝑘 = 𝐴))
2019simprd 497 . . . . . 6 (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝑘 = 𝐴)
2120eximi 1838 . . . . 5 (∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐴)
22 isset 3459 . . . . 5 (𝐴 ∈ V ↔ ∃𝑘 𝑘 = 𝐴)
2321, 22sylibr 233 . . . 4 (∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝐴 ∈ V)
249, 23syl 17 . . 3 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o𝐴 ∈ V)
25 1oex 8423 . . . . . . . 8 1o ∈ V
2625prid2 4725 . . . . . . 7 1o ∈ {∅, 1o}
2726, 4eleqtrri 2837 . . . . . 6 1o ∈ 2o
2827, 6eleqtrrid 2845 . . . . 5 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → 1o ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
2925eldm2 5858 . . . . 5 (1o ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ↔ ∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
3028, 29sylib 217 . . . 4 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
3110neii 2946 . . . . . . . . . 10 ¬ 1o = ∅
3225, 12opth1 5433 . . . . . . . . . 10 (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ → 1o = ∅)
3331, 32mto 196 . . . . . . . . 9 ¬ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴
34 elpri 4609 . . . . . . . . . 10 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩))
3534orcomd 870 . . . . . . . . 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 5434 . . . . . . . 8 (⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩ ↔ (1o = 1o𝑘 = 𝐵))
3937, 38sylib 217 . . . . . . 7 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (1o = 1o𝑘 = 𝐵))
4039simprd 497 . . . . . 6 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝑘 = 𝐵)
4140eximi 1838 . . . . 5 (∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐵)
42 isset 3459 . . . . 5 (𝐵 ∈ V ↔ ∃𝑘 𝑘 = 𝐵)
4341, 42sylibr 233 . . . 4 (∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝐵 ∈ V)
4430, 43syl 17 . . 3 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o𝐵 ∈ V)
4524, 44jca 513 . 2 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))
461, 45impbii 208 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397  wo 846   = wceq 1542  wex 1782  wcel 2107  Vcvv 3446  c0 4283  {cpr 4589  cop 4593  dom cdm 5634   Fn wfn 6492  1oc1o 8406  2oc2o 8407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-fun 6499  df-fn 6500  df-om 7804  df-1o 8413  df-2o 8414
This theorem is referenced by:  xpsfrnel2  17447
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