Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnpr2ob Structured version   Visualization version   GIF version

Theorem fnpr2ob 16833
 Description: Biconditional version of fnpr2o 16832. (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 16832 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
2 0ex 5213 . . . . . . . 8 ∅ ∈ V
32prid1 4700 . . . . . . 7 ∅ ∈ {∅, 1o}
4 df2o3 8119 . . . . . . 7 2o = {∅, 1o}
53, 4eleqtrri 2914 . . . . . 6 ∅ ∈ 2o
6 fndm 6457 . . . . . 6 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} = 2o)
75, 6eleqtrrid 2922 . . . . 5 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∅ ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
82eldm2 5772 . . . . 5 (∅ ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ↔ ∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
97, 8sylib 220 . . . 4 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
10 1n0 8121 . . . . . . . . . . 11 1o ≠ ∅
1110nesymi 3075 . . . . . . . . . 10 ¬ ∅ = 1o
12 vex 3499 . . . . . . . . . . 11 𝑘 ∈ V
132, 12opth1 5369 . . . . . . . . . 10 (⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩ → ∅ = 1o)
1411, 13mto 199 . . . . . . . . 9 ¬ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵
15 elpri 4591 . . . . . . . . 9 (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩))
16 orel2 887 . . . . . . . . 9 (¬ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩ → ((⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩) → ⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩))
1714, 15, 16mpsyl 68 . . . . . . . 8 (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩)
182, 12opth 5370 . . . . . . . 8 (⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ↔ (∅ = ∅ ∧ 𝑘 = 𝐴))
1917, 18sylib 220 . . . . . . 7 (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (∅ = ∅ ∧ 𝑘 = 𝐴))
2019simprd 498 . . . . . 6 (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝑘 = 𝐴)
2120eximi 1835 . . . . 5 (∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐴)
22 isset 3508 . . . . 5 (𝐴 ∈ V ↔ ∃𝑘 𝑘 = 𝐴)
2321, 22sylibr 236 . . . 4 (∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝐴 ∈ V)
249, 23syl 17 . . 3 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o𝐴 ∈ V)
25 1oex 8112 . . . . . . . 8 1o ∈ V
2625prid2 4701 . . . . . . 7 1o ∈ {∅, 1o}
2726, 4eleqtrri 2914 . . . . . 6 1o ∈ 2o
2827, 6eleqtrrid 2922 . . . . 5 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → 1o ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
2925eldm2 5772 . . . . 5 (1o ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ↔ ∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
3028, 29sylib 220 . . . 4 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
3110neii 3020 . . . . . . . . . 10 ¬ 1o = ∅
3225, 12opth1 5369 . . . . . . . . . 10 (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ → 1o = ∅)
3331, 32mto 199 . . . . . . . . 9 ¬ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴
34 elpri 4591 . . . . . . . . . 10 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩))
3534orcomd 867 . . . . . . . . 9 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩ ∨ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩))
36 orel2 887 . . . . . . . . 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 5370 . . . . . . . 8 (⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩ ↔ (1o = 1o𝑘 = 𝐵))
3937, 38sylib 220 . . . . . . 7 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (1o = 1o𝑘 = 𝐵))
4039simprd 498 . . . . . 6 (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝑘 = 𝐵)
4140eximi 1835 . . . . 5 (∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐵)
42 isset 3508 . . . . 5 (𝐵 ∈ V ↔ ∃𝑘 𝑘 = 𝐵)
4341, 42sylibr 236 . . . 4 (∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝐵 ∈ V)
4430, 43syl 17 . . 3 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o𝐵 ∈ V)
4524, 44jca 514 . 2 ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))
461, 45impbii 211 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3496  ∅c0 4293  {cpr 4571  ⟨cop 4575  dom cdm 5557   Fn wfn 6352  1oc1o 8097  2oc2o 8098 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-fun 6359  df-fn 6360  df-om 7583  df-1o 8104  df-2o 8105 This theorem is referenced by:  xpsfrnel2  16839
 Copyright terms: Public domain W3C validator