Theorem rrx2plordisom 48712

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 2729	. . . . 5 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})
3	1, 2	rrx2xpref1o 48707	. . . 4 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}):(ℝ × ℝ)–1-1-onto→𝑅
4		elxpi 5660	. . . . . 6 ⊢ (𝑎 ∈ (ℝ × ℝ) → ∃𝑐∃𝑑(𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)))
5		elxpi 5660	. . . . . 6 ⊢ (𝑏 ∈ (ℝ × ℝ) → ∃𝑒∃𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)))
6		df-br 5108	. . . . . . . . . . . . 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 5675	. . . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → 𝑎 ∈ (ℝ × ℝ))
12		opelxpi 5675	. . . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → 𝑏 ∈ (ℝ × ℝ))
17		fveq2 6858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (1st ‘𝑥) = (1st ‘𝑎))
18		fveq2 6858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → (1st ‘𝑦) = (1st ‘𝑏))
19	17, 18	breqan12d 5123	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((1st ‘𝑥) < (1st ‘𝑦) ↔ (1st ‘𝑎) < (1st ‘𝑏)))
20	17, 18	eqeqan12d 2743	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((1st ‘𝑥) = (1st ‘𝑦) ↔ (1st ‘𝑎) = (1st ‘𝑏)))
21		fveq2 6858	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (2nd ‘𝑥) = (2nd ‘𝑎))
22		fveq2 6858	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (2nd ‘𝑦) = (2nd ‘𝑏))
23	21, 22	breqan12d 5123	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((2nd ‘𝑥) < (2nd ‘𝑦) ↔ (2nd ‘𝑎) < (2nd ‘𝑏)))
24	20, 23	anbi12d 632	. . . . . . . . . . . . . . . 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 918	. . . . . . . . . . . . . . 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 5495	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 283	. . . . . . . . . . . 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 12389	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 2
30		1ex 11170	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
31		vex 3451	. . . . . . . . . . . . . . . . 17 ⊢ 𝑐 ∈ V
32	30, 31	fvpr1 7166	. . . . . . . . . . . . . . . 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 3451	. . . . . . . . . . . . . . . . 17 ⊢ 𝑒 ∈ V
35	30, 34	fvpr1 7166	. . . . . . . . . . . . . . . 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 5120	. . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . 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 12263	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ V
40		vex 3451	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑑 ∈ V
41	39, 40	fvpr2 7167	. . . . . . . . . . . . . . . . 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 3451	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
44	39, 43	fvpr2 7167	. . . . . . . . . . . . . . . . 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 5120	. . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 918	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ {1, 2} = {1, 2}
50	49, 1	prelrrx2 48702	. . . . . . . . . . . . . . 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 48702	. . . . . . . . . . . . . . 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 48709	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 7978	. . . . . . . . . . . . . . . 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 7978	. . . . . . . . . . . . . . . 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 5123	. . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . 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 7979	. . . . . . . . . . . . . . . . 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 7979	. . . . . . . . . . . . . . . . 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 5123	. . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 918	. . . . . . . . . . . . 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 312	. . . . . . . . . . . 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 6858	. . . . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑑) → ⟨1, 𝑥⟩ = ⟨1, 𝑐⟩)
77		opeq2 4838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑑 → ⟨2, 𝑦⟩ = ⟨2, 𝑑⟩)
78	77	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑑) → ⟨2, 𝑦⟩ = ⟨2, 𝑑⟩)
79	76, 78	preq12d 4705	. . . . . . . . . . . . . . . . 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 5392	. . . . . . . . . . . . . . . . 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 7541	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑐(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑑) = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
86	73, 85	sylan9eq 2784	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎) = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
87	86	eqcomd 2735	. . . . . . . . . . . . 13 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎))
88		fveq2 6858	. . . . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑒 ∧ 𝑦 = 𝑓) → ⟨1, 𝑥⟩ = ⟨1, 𝑒⟩)
94		opeq2 4838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑓 → ⟨2, 𝑦⟩ = ⟨2, 𝑓⟩)
95	94	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑒 ∧ 𝑦 = 𝑓) → ⟨2, 𝑦⟩ = ⟨2, 𝑓⟩)
96	93, 95	preq12d 4705	. . . . . . . . . . . . . . . . 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 5392	. . . . . . . . . . . . . . . . 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 7541	. . . . . . . . . . . . . . 15 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → (𝑒(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑓) = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
103	90, 102	sylan9eq 2784	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏) = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
104	103	eqcomd 2735	. . . . . . . . . . . . 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 5123	. . . . . . . . . . . 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 305	. . . . . . . . . . 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 1932	. . . . . . . . 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 1932	. . . . . . 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 596	. . . . 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 3177	. . . 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 6520	. . . 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 711	. . 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 7293	. . . 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 231	. 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 7292	. . 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 231	1 ⊢ 𝐹 Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3447  {cpr 4591  ⟨cop 4595   class class class wbr 5107  {copab 5169   × cxp 5636  –1-1-onto→wf1o 6510  ‘cfv 6511   Isom wiso 6512  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967   ↑_m cmap 8799  ℝcr 11067  1c1 11069   < clt 11208  2c2 12241

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-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-2 12249

This theorem is referenced by:  rrx2plordso  48713