Theorem fnpr2ob 13042

Description: Biconditional version of fnpr2o 13041. (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 13041	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
2		0ex 4161	. . . . . . . 8 ⊢ ∅ ∈ V
3	2	prid1 3729	. . . . . . 7 ⊢ ∅ ∈ {∅, 1o}
4		df2o3 6497	. . . . . . 7 ⊢ 2o = {∅, 1o}
5	3, 4	eleqtrri 2272	. . . . . 6 ⊢ ∅ ∈ 2o
6		fndm 5358	. . . . . 6 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} = 2o)
7	5, 6	eleqtrrid 2286	. . . . 5 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∅ ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
8	2	eldm2 4865	. . . . 5 ⊢ (∅ ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ↔ ∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
9	7, 8	sylib 122	. . . 4 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
10		1n0 6499	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
11	10	nesymi 2413	. . . . . . . . . 10 ⊢ ¬ ∅ = 1o
12		vex 2766	. . . . . . . . . . 11 ⊢ 𝑘 ∈ V
13	2, 12	opth1 4270	. . . . . . . . . 10 ⊢ (⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩ → ∅ = 1o)
14	11, 13	mto 663	. . . . . . . . 9 ⊢ ¬ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩
15		elpri 3646	. . . . . . . . 9 ⊢ (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩))
16		orel2 727	. . . . . . . . 9 ⊢ (¬ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩ → ((⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨∅, 𝑘⟩ = ⟨1o, 𝐵⟩) → ⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩))
17	14, 15, 16	mpsyl 65	. . . . . . . 8 ⊢ (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩)
18	2, 12	opth 4271	. . . . . . . 8 ⊢ (⟨∅, 𝑘⟩ = ⟨∅, 𝐴⟩ ↔ (∅ = ∅ ∧ 𝑘 = 𝐴))
19	17, 18	sylib 122	. . . . . . 7 ⊢ (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (∅ = ∅ ∧ 𝑘 = 𝐴))
20	19	simprd 114	. . . . . 6 ⊢ (⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝑘 = 𝐴)
21	20	eximi 1614	. . . . 5 ⊢ (∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐴)
22		isset 2769	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑘 𝑘 = 𝐴)
23	21, 22	sylibr 134	. . . 4 ⊢ (∃𝑘⟨∅, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝐴 ∈ V)
24	9, 23	syl 14	. . 3 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → 𝐴 ∈ V)
25		1oex 6491	. . . . . . . 8 ⊢ 1o ∈ V
26	25	prid2 3730	. . . . . . 7 ⊢ 1o ∈ {∅, 1o}
27	26, 4	eleqtrri 2272	. . . . . 6 ⊢ 1o ∈ 2o
28	27, 6	eleqtrrid 2286	. . . . 5 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → 1o ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
29	25	eldm2 4865	. . . . 5 ⊢ (1o ∈ dom {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ↔ ∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
30	28, 29	sylib 122	. . . 4 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → ∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
31	10	neii 2369	. . . . . . . . . 10 ⊢ ¬ 1o = ∅
32	25, 12	opth1 4270	. . . . . . . . . 10 ⊢ (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ → 1o = ∅)
33	31, 32	mto 663	. . . . . . . . 9 ⊢ ¬ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩
34		elpri 3646	. . . . . . . . . 10 ⊢ (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ ∨ ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩))
35	34	orcomd 730	. . . . . . . . 9 ⊢ (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩ ∨ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩))
36		orel2 727	. . . . . . . . 9 ⊢ (¬ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩ → ((⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩ ∨ ⟨1o, 𝑘⟩ = ⟨∅, 𝐴⟩) → ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩))
37	33, 35, 36	mpsyl 65	. . . . . . . 8 ⊢ (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩)
38	25, 12	opth 4271	. . . . . . . 8 ⊢ (⟨1o, 𝑘⟩ = ⟨1o, 𝐵⟩ ↔ (1o = 1o ∧ 𝑘 = 𝐵))
39	37, 38	sylib 122	. . . . . . 7 ⊢ (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (1o = 1o ∧ 𝑘 = 𝐵))
40	39	simprd 114	. . . . . 6 ⊢ (⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝑘 = 𝐵)
41	40	eximi 1614	. . . . 5 ⊢ (∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → ∃𝑘 𝑘 = 𝐵)
42		isset 2769	. . . . 5 ⊢ (𝐵 ∈ V ↔ ∃𝑘 𝑘 = 𝐵)
43	41, 42	sylibr 134	. . . 4 ⊢ (∃𝑘⟨1o, 𝑘⟩ ∈ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → 𝐵 ∈ V)
44	30, 43	syl 14	. . 3 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → 𝐵 ∈ V)
45	24, 44	jca 306	. 2 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))
46	1, 45	impbii 126	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763  ∅c0 3451  {cpr 3624  ⟨cop 3626  dom cdm 4664   Fn wfn 5254  1_oc1o 6476  2_oc2o 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-fun 5261  df-fn 5262  df-1o 6483  df-2o 6484

This theorem is referenced by:  xpsfrnel2  13048