Theorem fnpr2ob 17480

Description: Biconditional version of fnpr2o 17479. (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 17479	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
2		0ex 5249	. . . . . . . 8 ⊢ ∅ ∈ V
3	2	prid1 4716	. . . . . . 7 ⊢ ∅ ∈ {∅, 1o}
4		df2o3 8403	. . . . . . 7 ⊢ 2o = {∅, 1o}
5	3, 4	eleqtrri 2827	. . . . . 6 ⊢ ∅ ∈ 2o
6		fndm 6589	. . . . . 6 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} = 2o)
7	5, 6	eleqtrrid 2835	. . . . 5 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∅ ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
8	2	eldm2 5848	. . . . 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 8413	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
11	10	nesymi 2982	. . . . . . . . . 10 ⊢ ¬ ∅ = 1o
12		vex 3442	. . . . . . . . . . 11 ⊢ 𝑘 ∈ V
13	2, 12	opth1 5422	. . . . . . . . . 10 ⊢ (⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩ → ∅ = 1o)
14	11, 13	mto 197	. . . . . . . . 9 ⊢ ¬ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩
15		elpri 4603	. . . . . . . . 9 ⊢ (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩))
16		orel2 890	. . . . . . . . 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 5423	. . . . . . . 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 1835	. . . . 5 ⊢ (∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐴)
22		isset 3452	. . . . 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 8405	. . . . . . . 8 ⊢ 1o ∈ V
26	25	prid2 4717	. . . . . . 7 ⊢ 1o ∈ {∅, 1o}
27	26, 4	eleqtrri 2827	. . . . . 6 ⊢ 1o ∈ 2o
28	27, 6	eleqtrrid 2835	. . . . 5 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → 1o ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
29	25	eldm2 5848	. . . . 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 2927	. . . . . . . . . 10 ⊢ ¬ 1o = ∅
32	25, 12	opth1 5422	. . . . . . . . . 10 ⊢ (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ → 1o = ∅)
33	31, 32	mto 197	. . . . . . . . 9 ⊢ ¬ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩
34		elpri 4603	. . . . . . . . . 10 ⊢ (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩))
35	34	orcomd 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⟩))
37	33, 35, 36	mpsyl 68	. . . . . . . 8 ⊢ (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩)
38	25, 12	opth 5423	. . . . . . . 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 1835	. . . . 5 ⊢ (∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐵)
42		isset 3452	. . . . 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 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3438  ∅c0 4286  {cpr 4581  ⟨cop 4585  dom cdm 5623   Fn wfn 6481  1_oc1o 8388  2_oc2o 8389

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-fun 6488  df-fn 6489  df-om 7807  df-1o 8395  df-2o 8396

This theorem is referenced by:  xpsfrnel2  17486