Theorem rrx2plordisom 47498

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 2731	. . . . 5 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})
3	1, 2	rrx2xpref1o 47493	. . . 4 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}):(ℝ × ℝ)–1-1-onto→𝑅
4		elxpi 5699	. . . . . 6 ⊢ (𝑎 ∈ (ℝ × ℝ) → ∃𝑐∃𝑑(𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)))
5		elxpi 5699	. . . . . 6 ⊢ (𝑏 ∈ (ℝ × ℝ) → ∃𝑒∃𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)))
6		df-br 5150	. . . . . . . . . . . . 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 5714	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → ⟨𝑐, 𝑑⟩ ∈ (ℝ × ℝ))
8	7	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ⟨𝑐, 𝑑⟩ ∈ (ℝ × ℝ))
9		eleq1 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ⟨𝑐, 𝑑⟩ → (𝑎 ∈ (ℝ × ℝ) ↔ ⟨𝑐, 𝑑⟩ ∈ (ℝ × ℝ)))
10	9	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (𝑎 ∈ (ℝ × ℝ) ↔ ⟨𝑐, 𝑑⟩ ∈ (ℝ × ℝ)))
11	8, 10	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → 𝑎 ∈ (ℝ × ℝ))
12		opelxpi 5714	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → ⟨𝑒, 𝑓⟩ ∈ (ℝ × ℝ))
13	12	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → ⟨𝑒, 𝑓⟩ ∈ (ℝ × ℝ))
14		eleq1 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = ⟨𝑒, 𝑓⟩ → (𝑏 ∈ (ℝ × ℝ) ↔ ⟨𝑒, 𝑓⟩ ∈ (ℝ × ℝ)))
15	14	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (𝑏 ∈ (ℝ × ℝ) ↔ ⟨𝑒, 𝑓⟩ ∈ (ℝ × ℝ)))
16	13, 15	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → 𝑏 ∈ (ℝ × ℝ))
17		fveq2 6892	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (1st ‘𝑥) = (1st ‘𝑎))
18		fveq2 6892	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → (1st ‘𝑦) = (1st ‘𝑏))
19	17, 18	breqan12d 5165	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((1st ‘𝑥) < (1st ‘𝑦) ↔ (1st ‘𝑎) < (1st ‘𝑏)))
20	17, 18	eqeqan12d 2745	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((1st ‘𝑥) = (1st ‘𝑦) ↔ (1st ‘𝑎) = (1st ‘𝑏)))
21		fveq2 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (2nd ‘𝑥) = (2nd ‘𝑎))
22		fveq2 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (2nd ‘𝑦) = (2nd ‘𝑏))
23	21, 22	breqan12d 5165	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((2nd ‘𝑥) < (2nd ‘𝑦) ↔ (2nd ‘𝑎) < (2nd ‘𝑏)))
24	20, 23	anbi12d 630	. . . . . . . . . . . . . . . 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 916	. . . . . . . . . . . . . . 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 5536	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 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 282	. . . . . . . . . . . 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 12425	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 2
30		1ex 11215	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
31		vex 3477	. . . . . . . . . . . . . . . . 17 ⊢ 𝑐 ∈ V
32	30, 31	fvpr1 7194	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 2 → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) = 𝑐)
33	29, 32	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) = 𝑐)
34		vex 3477	. . . . . . . . . . . . . . . . 17 ⊢ 𝑒 ∈ V
35	30, 34	fvpr1 7194	. . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 2 → ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) = 𝑒)
36	29, 35	mp1i 13	. . . . . . . . . . . . . . 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 5162	. . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . . 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 12294	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ V
40		vex 3477	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑑 ∈ V
41	39, 40	fvpr2 7196	. . . . . . . . . . . . . . . . 17 ⊢ (1 ≠ 2 → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘2) = 𝑑)
42	29, 41	mp1i 13	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘2) = 𝑑)
43		vex 3477	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
44	39, 43	fvpr2 7196	. . . . . . . . . . . . . . . . 17 ⊢ (1 ≠ 2 → ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘2) = 𝑓)
45	29, 44	mp1i 13	. . . . . . . . . . . . . . . 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 5162	. . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . 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 916	. . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . 16 ⊢ {1, 2} = {1, 2}
50	49, 1	prelrrx2 47488	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} ∈ 𝑅)
51	50	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} ∈ 𝑅)
52	49, 1	prelrrx2 47488	. . . . . . . . . . . . . . 15 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → {⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ∈ 𝑅)
53	52	adantl 481	. . . . . . . . . . . . . 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 47495	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 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 7988	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ⟨𝑐, 𝑑⟩ → (1st ‘𝑎) = 𝑐)
58	57	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (1st ‘𝑎) = 𝑐)
59	34, 43	op1std 7988	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = ⟨𝑒, 𝑓⟩ → (1st ‘𝑏) = 𝑒)
60	59	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (1st ‘𝑏) = 𝑒)
61	58, 60	breqan12d 5165	. . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . 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 7989	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = ⟨𝑐, 𝑑⟩ → (2nd ‘𝑎) = 𝑑)
64	63	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (2nd ‘𝑎) = 𝑑)
65	34, 43	op2ndd 7989	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = ⟨𝑒, 𝑓⟩ → (2nd ‘𝑏) = 𝑓)
66	65	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (2nd ‘𝑏) = 𝑓)
67	64, 66	breqan12d 5165	. . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . 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 916	. . . . . . . . . . . . 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 311	. . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}))
75		opeq2 4875	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑐 → ⟨1, 𝑥⟩ = ⟨1, 𝑐⟩)
76	75	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑑) → ⟨1, 𝑥⟩ = ⟨1, 𝑐⟩)
77		opeq2 4875	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑑 → ⟨2, 𝑦⟩ = ⟨2, 𝑑⟩)
78	77	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑑) → ⟨2, 𝑦⟩ = ⟨2, 𝑑⟩)
79	76, 78	preq12d 4746	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑑) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
80	79	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) ∧ (𝑥 = 𝑐 ∧ 𝑦 = 𝑑)) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
81		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → 𝑐 ∈ ℝ)
82		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → 𝑑 ∈ ℝ)
83		prex 5433	. . . . . . . . . . . . . . . . 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 7563	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑐(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑑) = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
86	73, 85	sylan9eq 2791	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎) = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
87	86	eqcomd 2737	. . . . . . . . . . . . 13 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎))
88		fveq2 6892	. . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}))
92		opeq2 4875	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑒 → ⟨1, 𝑥⟩ = ⟨1, 𝑒⟩)
93	92	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑒 ∧ 𝑦 = 𝑓) → ⟨1, 𝑥⟩ = ⟨1, 𝑒⟩)
94		opeq2 4875	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑓 → ⟨2, 𝑦⟩ = ⟨2, 𝑓⟩)
95	94	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑒 ∧ 𝑦 = 𝑓) → ⟨2, 𝑦⟩ = ⟨2, 𝑓⟩)
96	93, 95	preq12d 4746	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑒 ∧ 𝑦 = 𝑓) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
97	96	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) ∧ (𝑥 = 𝑒 ∧ 𝑦 = 𝑓)) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
98		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → 𝑒 ∈ ℝ)
99		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → 𝑓 ∈ ℝ)
100		prex 5433	. . . . . . . . . . . . . . . . 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 7563	. . . . . . . . . . . . . . 15 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → (𝑒(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑓) = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
103	90, 102	sylan9eq 2791	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏) = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
104	103	eqcomd 2737	. . . . . . . . . . . . 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 5165	. . . . . . . . . . . 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 304	. . . . . . . . . . 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 413	. . . . . . . . . 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 1934	. . . . . . . . 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 32	. . . . . . . 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 1934	. . . . . . 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 406	. . . . . 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 595	. . . . 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 3196	. . . 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 6553	. . . 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 708	. . 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 7318	. . . 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 230	. 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 7317	. . 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 230	1 ⊢ 𝐹 Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1540  ∃wex 1780   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  Vcvv 3473  {cpr 4631  ⟨cop 4635   class class class wbr 5149  {copab 5211   × cxp 5675  –1-1-onto→wf1o 6543  ‘cfv 6544   Isom wiso 6545  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7976  2^nd c2nd 7977   ↑_m cmap 8823  ℝcr 11112  1c1 11114   < clt 11253  2c2 12272

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-er 8706  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-2 12280

This theorem is referenced by:  rrx2plordso  47499