Theorem fnpr2ob 17513

Description: Biconditional version of fnpr2o 17512. (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.

Step	Hyp	Ref	Expression
1		fnpr2o 17512	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
2		0ex 5242	. . . . . . . 8 ⊢ ∅ ∈ V
3	2	prid1 4707	. . . . . . 7 ⊢ ∅ ∈ {∅, 1o}
4		df2o3 8406	. . . . . . 7 ⊢ 2o = {∅, 1o}
5	3, 4	eleqtrri 2836	. . . . . 6 ⊢ ∅ ∈ 2o
6		fndm 6595	. . . . . 6 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} = 2o)
7	5, 6	eleqtrrid 2844	. . . . 5 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∅ ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
8	2	eldm2 5850	. . . . 5 ⊢ (∅ ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ↔ ∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
9	7, 8	sylib 218	. . . 4 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
10		1n0 8416	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
11	10	nesymi 2990	. . . . . . . . . 10 ⊢ ¬ ∅ = 1o
12		vex 3434	. . . . . . . . . . 11 ⊢ 𝑘 ∈ V
13	2, 12	opth1 5423	. . . . . . . . . 10 ⊢ (⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩ → ∅ = 1o)
14	11, 13	mto 197	. . . . . . . . 9 ⊢ ¬ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩
15		elpri 4592	. . . . . . . . 9 ⊢ (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩))
16		orel2 891	. . . . . . . . 9 ⊢ (¬ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩ → ((⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩) → ⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩))
17	14, 15, 16	mpsyl 68	. . . . . . . 8 ⊢ (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩)
18	2, 12	opth 5424	. . . . . . . 8 ⊢ (⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ↔ (∅ = ∅ ∧ 𝑘 = 𝐴))
19	17, 18	sylib 218	. . . . . . 7 ⊢ (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (∅ = ∅ ∧ 𝑘 = 𝐴))
20	19	simprd 495	. . . . . 6 ⊢ (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝑘 = 𝐴)
21	20	eximi 1837	. . . . 5 ⊢ (∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐴)
22		isset 3444	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑘 𝑘 = 𝐴)
23	21, 22	sylibr 234	. . . 4 ⊢ (∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝐴 ∈ V)
24	9, 23	syl 17	. . 3 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → 𝐴 ∈ V)
25		1oex 8408	. . . . . . . 8 ⊢ 1o ∈ V
26	25	prid2 4708	. . . . . . 7 ⊢ 1o ∈ {∅, 1o}
27	26, 4	eleqtrri 2836	. . . . . 6 ⊢ 1o ∈ 2o
28	27, 6	eleqtrrid 2844	. . . . 5 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → 1o ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
29	25	eldm2 5850	. . . . 5 ⊢ (1o ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ↔ ∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
30	28, 29	sylib 218	. . . 4 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
31	10	neii 2935	. . . . . . . . . 10 ⊢ ¬ 1o = ∅
32	25, 12	opth1 5423	. . . . . . . . . 10 ⊢ (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ → 1o = ∅)
33	31, 32	mto 197	. . . . . . . . 9 ⊢ ¬ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩
34		elpri 4592	. . . . . . . . . 10 ⊢ (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩))
35	34	orcomd 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⟩))
37	33, 35, 36	mpsyl 68	. . . . . . . 8 ⊢ (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩)
38	25, 12	opth 5424	. . . . . . . 8 ⊢ (⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩ ↔ (1o = 1o ∧ 𝑘 = 𝐵))
39	37, 38	sylib 218	. . . . . . 7 ⊢ (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (1o = 1o ∧ 𝑘 = 𝐵))
40	39	simprd 495	. . . . . 6 ⊢ (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝑘 = 𝐵)
41	40	eximi 1837	. . . . 5 ⊢ (∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐵)
42		isset 3444	. . . . 5 ⊢ (𝐵 ∈ V ↔ ∃𝑘 𝑘 = 𝐵)
43	41, 42	sylibr 234	. . . 4 ⊢ (∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝐵 ∈ V)
44	30, 43	syl 17	. . 3 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → 𝐵 ∈ V)
45	24, 44	jca 511	. 2 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))
46	1, 45	impbii 209	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  {cpr 4570  ⟨cop 4574  dom cdm 5624   Fn wfn 6487  1_oc1o 8391  2_oc2o 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-fun 6494  df-fn 6495  df-om 7811  df-1o 8398  df-2o 8399

This theorem is referenced by:  xpsfrnel2  17519