Theorem rrx2plordisom 49531

Description: The set of points in the two dimensional Euclidean plane with the lexicographical ordering is isomorphic to the cartesian product of the real numbers with the lexicographical ordering implied by the ordering of the real numbers. (Contributed by AV, 12-Mar-2023.)

Hypotheses

Ref	Expression
rrx2plord.o	⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}
rrx2plord2.r	⊢ 𝑅 = (ℝ ↑m {1, 2})
rrx2plordisom.f	⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})
rrx2plordisom.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}

Assertion

Ref	Expression
rrx2plordisom	⊢ 𝐹 Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅)

Distinct variable group:   𝑥,𝑅,𝑦

Allowed substitution hints:   𝑇(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝑂(𝑥, 𝑦)

Proof of Theorem rrx2plordisom

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrx2plord2.r	. . . . 5 ⊢ 𝑅 = (ℝ ↑m {1, 2})
2		eqid 2762	. . . . 5 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})
3	1, 2	rrx2xpref1o 49526	. . . 4 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}):(ℝ × ℝ)–1-1-onto→𝑅
4		elxpi 5682	. . . . . 6 ⊢ (𝑎 ∈ (ℝ × ℝ) → ∃𝑐∃𝑑(𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)))
5		elxpi 5682	. . . . . 6 ⊢ (𝑏 ∈ (ℝ × ℝ) → ∃𝑒∃𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)))
6		df-br 5109	. . . . . . . . . . . . 13 ⊢ (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))})
7		opelxpi 5697	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → ⟨𝑐, 𝑑⟩ ∈ (ℝ × ℝ))
8	7	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ⟨𝑐, 𝑑⟩ ∈ (ℝ × ℝ))
9		eleq1 2850	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ⟨𝑐, 𝑑⟩ → (𝑎 ∈ (ℝ × ℝ) ↔ ⟨𝑐, 𝑑⟩ ∈ (ℝ × ℝ)))
10	9	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (𝑎 ∈ (ℝ × ℝ) ↔ ⟨𝑐, 𝑑⟩ ∈ (ℝ × ℝ)))
11	8, 10	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → 𝑎 ∈ (ℝ × ℝ))
12		opelxpi 5697	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → ⟨𝑒, 𝑓⟩ ∈ (ℝ × ℝ))
13	12	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → ⟨𝑒, 𝑓⟩ ∈ (ℝ × ℝ))
14		eleq1 2850	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = ⟨𝑒, 𝑓⟩ → (𝑏 ∈ (ℝ × ℝ) ↔ ⟨𝑒, 𝑓⟩ ∈ (ℝ × ℝ)))
15	14	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (𝑏 ∈ (ℝ × ℝ) ↔ ⟨𝑒, 𝑓⟩ ∈ (ℝ × ℝ)))
16	13, 15	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → 𝑏 ∈ (ℝ × ℝ))
17		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (1st ‘𝑥) = (1st ‘𝑎))
18		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → (1st ‘𝑦) = (1st ‘𝑏))
19	17, 18	breqan12d 5124	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((1st ‘𝑥) < (1st ‘𝑦) ↔ (1st ‘𝑎) < (1st ‘𝑏)))
20	17, 18	eqeqan12d 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((1st ‘𝑥) = (1st ‘𝑦) ↔ (1st ‘𝑎) = (1st ‘𝑏)))
21		fveq2 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (2nd ‘𝑥) = (2nd ‘𝑎))
22		fveq2 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (2nd ‘𝑦) = (2nd ‘𝑏))
23	21, 22	breqan12d 5124	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((2nd ‘𝑥) < (2nd ‘𝑦) ↔ (2nd ‘𝑎) < (2nd ‘𝑏)))
24	20, 23	anbi12d 643	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦)) ↔ ((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) < (2nd ‘𝑏))))
25	19, 24	orbi12d 931	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))) ↔ ((1st ‘𝑎) < (1st ‘𝑏) ∨ ((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) < (2nd ‘𝑏)))))
26	25	opelopab2a 5518	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ (ℝ × ℝ) ∧ 𝑏 ∈ (ℝ × ℝ)) → (⟨𝑎, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))} ↔ ((1st ‘𝑎) < (1st ‘𝑏) ∨ ((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) < (2nd ‘𝑏)))))
27	11, 16, 26	syl2an 607	. . . . . . . . . . . . 13 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (⟨𝑎, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))} ↔ ((1st ‘𝑎) < (1st ‘𝑏) ∨ ((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) < (2nd ‘𝑏)))))
28	6, 27	bitrid 286	. . . . . . . . . . . 12 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}𝑏 ↔ ((1st ‘𝑎) < (1st ‘𝑏) ∨ ((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) < (2nd ‘𝑏)))))
29		1ne2 12457	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 2
30		1ex 11209	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
31		vex 3458	. . . . . . . . . . . . . . . . 17 ⊢ 𝑐 ∈ V
32	30, 31	fvpr1 7190	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 2 → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) = 𝑐)
33	29, 32	mp1i 14	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) = 𝑐)
34		vex 3458	. . . . . . . . . . . . . . . . 17 ⊢ 𝑒 ∈ V
35	30, 34	fvpr1 7190	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 2 → ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) = 𝑒)
36	29, 35	mp1i 14	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) = 𝑒)
37	33, 36	breq12d 5121	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) < ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) ↔ 𝑐 < 𝑒))
38	33, 36	eqeq12d 2778	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) = ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) ↔ 𝑐 = 𝑒))
39		2ex 12324	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ V
40		vex 3458	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑑 ∈ V
41	39, 40	fvpr2 7191	. . . . . . . . . . . . . . . . 17 ⊢ (1 ≠ 2 → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘2) = 𝑑)
42	29, 41	mp1i 14	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘2) = 𝑑)
43		vex 3458	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
44	39, 43	fvpr2 7191	. . . . . . . . . . . . . . . . 17 ⊢ (1 ≠ 2 → ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘2) = 𝑓)
45	29, 44	mp1i 14	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘2) = 𝑓)
46	42, 45	breq12d 5121	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘2) < ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘2) ↔ 𝑑 < 𝑓))
47	38, 46	anbi12d 643	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ((({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) = ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) ∧ ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘2) < ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘2)) ↔ (𝑐 = 𝑒 ∧ 𝑑 < 𝑓)))
48	37, 47	orbi12d 931	. . . . . . . . . . . . 13 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ((({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) < ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) ∨ (({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) = ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) ∧ ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘2) < ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘2))) ↔ (𝑐 < 𝑒 ∨ (𝑐 = 𝑒 ∧ 𝑑 < 𝑓))))
49		eqid 2762	. . . . . . . . . . . . . . . 16 ⊢ {1, 2} = {1, 2}
50	49, 1	prelrrx2 49521	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} ∈ 𝑅)
51	50	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} ∈ 𝑅)
52	49, 1	prelrrx2 49521	. . . . . . . . . . . . . . 15 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → {⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ∈ 𝑅)
53	52	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → {⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ∈ 𝑅)
54		rrx2plord.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}
55	54	rrx2plord 49528	. . . . . . . . . . . . . 14 ⊢ (({⟨1, 𝑐⟩, ⟨2, 𝑑⟩} ∈ 𝑅 ∧ {⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ∈ 𝑅) → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}𝑂{⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ↔ (({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) < ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) ∨ (({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) = ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) ∧ ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘2) < ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘2)))))
56	51, 53, 55	syl2an 607	. . . . . . . . . . . . 13 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}𝑂{⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ↔ (({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) < ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) ∨ (({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) = ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) ∧ ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘2) < ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘2)))))
57	31, 40	op1std 7994	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ⟨𝑐, 𝑑⟩ → (1st ‘𝑎) = 𝑐)
58	57	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (1st ‘𝑎) = 𝑐)
59	34, 43	op1std 7994	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = ⟨𝑒, 𝑓⟩ → (1st ‘𝑏) = 𝑒)
60	59	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (1st ‘𝑏) = 𝑒)
61	58, 60	breqan12d 5124	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ((1st ‘𝑎) < (1st ‘𝑏) ↔ 𝑐 < 𝑒))
62	58, 60	eqeqan12d 2776	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ((1st ‘𝑎) = (1st ‘𝑏) ↔ 𝑐 = 𝑒))
63	31, 40	op2ndd 7995	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = ⟨𝑐, 𝑑⟩ → (2nd ‘𝑎) = 𝑑)
64	63	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (2nd ‘𝑎) = 𝑑)
65	34, 43	op2ndd 7995	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = ⟨𝑒, 𝑓⟩ → (2nd ‘𝑏) = 𝑓)
66	65	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (2nd ‘𝑏) = 𝑓)
67	64, 66	breqan12d 5124	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ((2nd ‘𝑎) < (2nd ‘𝑏) ↔ 𝑑 < 𝑓))
68	62, 67	anbi12d 643	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) < (2nd ‘𝑏)) ↔ (𝑐 = 𝑒 ∧ 𝑑 < 𝑓)))
69	61, 68	orbi12d 931	. . . . . . . . . . . . 13 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (((1st ‘𝑎) < (1st ‘𝑏) ∨ ((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) < (2nd ‘𝑏))) ↔ (𝑐 < 𝑒 ∨ (𝑐 = 𝑒 ∧ 𝑑 < 𝑓))))
70	48, 56, 69	3bitr4rd 315	. . . . . . . . . . . 12 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (((1st ‘𝑎) < (1st ‘𝑏) ∨ ((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) < (2nd ‘𝑏))) ↔ {⟨1, 𝑐⟩, ⟨2, 𝑑⟩}𝑂{⟨1, 𝑒⟩, ⟨2, 𝑓⟩}))
71		fveq2 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ⟨𝑐, 𝑑⟩ → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎) = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘⟨𝑐, 𝑑⟩))
72		df-ov 7415	. . . . . . . . . . . . . . . 16 ⊢ (𝑐(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑑) = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘⟨𝑐, 𝑑⟩)
73	71, 72	eqtr4di 2815	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ⟨𝑐, 𝑑⟩ → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎) = (𝑐(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑑))
74		eqidd 2763	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}))
75		opeq2 4838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑐 → ⟨1, 𝑥⟩ = ⟨1, 𝑐⟩)
76	75	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑑) → ⟨1, 𝑥⟩ = ⟨1, 𝑐⟩)
77		opeq2 4838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑑 → ⟨2, 𝑦⟩ = ⟨2, 𝑑⟩)
78	77	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑑) → ⟨2, 𝑦⟩ = ⟨2, 𝑑⟩)
79	76, 78	preq12d 4706	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑑) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
80	79	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) ∧ (𝑥 = 𝑐 ∧ 𝑦 = 𝑑)) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
81		simpl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → 𝑐 ∈ ℝ)
82		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → 𝑑 ∈ ℝ)
83		prex 5408	. . . . . . . . . . . . . . . . 17 ⊢ {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} ∈ V
84	83	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} ∈ V)
85	74, 80, 81, 82, 84	ovmpod 7564	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑐(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑑) = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
86	73, 85	sylan9eq 2817	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎) = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
87	86	eqcomd 2768	. . . . . . . . . . . . 13 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎))
88		fveq2 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = ⟨𝑒, 𝑓⟩ → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏) = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘⟨𝑒, 𝑓⟩))
89		df-ov 7415	. . . . . . . . . . . . . . . 16 ⊢ (𝑒(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑓) = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘⟨𝑒, 𝑓⟩)
90	88, 89	eqtr4di 2815	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ⟨𝑒, 𝑓⟩ → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏) = (𝑒(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑓))
91		eqidd 2763	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}))
92		opeq2 4838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑒 → ⟨1, 𝑥⟩ = ⟨1, 𝑒⟩)
93	92	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑒 ∧ 𝑦 = 𝑓) → ⟨1, 𝑥⟩ = ⟨1, 𝑒⟩)
94		opeq2 4838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑓 → ⟨2, 𝑦⟩ = ⟨2, 𝑓⟩)
95	94	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑒 ∧ 𝑦 = 𝑓) → ⟨2, 𝑦⟩ = ⟨2, 𝑓⟩)
96	93, 95	preq12d 4706	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑒 ∧ 𝑦 = 𝑓) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
97	96	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) ∧ (𝑥 = 𝑒 ∧ 𝑦 = 𝑓)) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
98		simpl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → 𝑒 ∈ ℝ)
99		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → 𝑓 ∈ ℝ)
100		prex 5408	. . . . . . . . . . . . . . . . 17 ⊢ {⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ∈ V
101	100	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → {⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ∈ V)
102	91, 97, 98, 99, 101	ovmpod 7564	. . . . . . . . . . . . . . 15 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → (𝑒(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑓) = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
103	90, 102	sylan9eq 2817	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏) = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
104	103	eqcomd 2768	. . . . . . . . . . . . 13 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → {⟨1, 𝑒⟩, ⟨2, 𝑓⟩} = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))
105	87, 104	breqan12d 5124	. . . . . . . . . . . 12 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}𝑂{⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏)))
106	28, 70, 105	3bitrd 308	. . . . . . . . . . 11 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏)))
107	106	expcom 418	. . . . . . . . . 10 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))))
108	107	exlimivv 1961	. . . . . . . . 9 ⊢ (∃𝑒∃𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))))
109	108	com12 33	. . . . . . . 8 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (∃𝑒∃𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))))
110	109	exlimivv 1961	. . . . . . 7 ⊢ (∃𝑐∃𝑑(𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (∃𝑒∃𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))))
111	110	imp 411	. . . . . 6 ⊢ ((∃𝑐∃𝑑(𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ ∃𝑒∃𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏)))
112	4, 5, 111	syl2an 607	. . . . 5 ⊢ ((𝑎 ∈ (ℝ × ℝ) ∧ 𝑏 ∈ (ℝ × ℝ)) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏)))
113	112	rgen2 3204	. . . 4 ⊢ ∀𝑎 ∈ (ℝ × ℝ)∀𝑏 ∈ (ℝ × ℝ)(𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))
114		df-isom 6545	. . . 4 ⊢ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}, 𝑂((ℝ × ℝ), 𝑅) ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}):(ℝ × ℝ)–1-1-onto→𝑅 ∧ ∀𝑎 ∈ (ℝ × ℝ)∀𝑏 ∈ (ℝ × ℝ)(𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))))
115	3, 113, 114	mpbir2an 723	. . 3 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}, 𝑂((ℝ × ℝ), 𝑅)
116		rrx2plordisom.t	. . . 4 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}
117		isoeq2 7316	. . . 4 ⊢ (𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))} → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅) ↔ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}, 𝑂((ℝ × ℝ), 𝑅)))
118	116, 117	ax-mp 5	. . 3 ⊢ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅) ↔ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}, 𝑂((ℝ × ℝ), 𝑅))
119	115, 118	mpbir 234	. 2 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅)
120		rrx2plordisom.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})
121		isoeq1 7315	. . 3 ⊢ (𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) → (𝐹 Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅) ↔ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅)))
122	120, 121	ax-mp 5	. 2 ⊢ (𝐹 Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅) ↔ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅))
123	119, 122	mpbir 234	1 ⊢ 𝐹 Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1569  ∃wex 1808   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  Vcvv 3454  {cpr 4590  ⟨cop 4594   class class class wbr 5108  {copab 5172   × cxp 5658  –1-1-onto→wf1o 6535  ‘cfv 6536   Isom wiso 6537  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7982  2^nd c2nd 7983   ↑_m cmap 8822  ℝcr 11105  1c1 11107   < clt 11249  2c2 12301

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-po 5568  df-so 5569  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-er 8692  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-2 12309

This theorem is used by:  rrx2plordso  49532