Theorem rrx2plordisom 48644

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 2737	. . . . 5 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})
3	1, 2	rrx2xpref1o 48639	. . . 4 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}):(ℝ × ℝ)–1-1-onto→𝑅
4		elxpi 5707	. . . . . 6 ⊢ (𝑎 ∈ (ℝ × ℝ) → ∃𝑐∃𝑑(𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)))
5		elxpi 5707	. . . . . 6 ⊢ (𝑏 ∈ (ℝ × ℝ) → ∃𝑒∃𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)))
6		df-br 5144	. . . . . . . . . . . . 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 5722	. . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . 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 5722	. . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (1st ‘𝑥) = (1st ‘𝑎))
18		fveq2 6906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → (1st ‘𝑦) = (1st ‘𝑏))
19	17, 18	breqan12d 5159	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((1st ‘𝑥) < (1st ‘𝑦) ↔ (1st ‘𝑎) < (1st ‘𝑏)))
20	17, 18	eqeqan12d 2751	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((1st ‘𝑥) = (1st ‘𝑦) ↔ (1st ‘𝑎) = (1st ‘𝑏)))
21		fveq2 6906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (2nd ‘𝑥) = (2nd ‘𝑎))
22		fveq2 6906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (2nd ‘𝑦) = (2nd ‘𝑏))
23	21, 22	breqan12d 5159	. . . . . . . . . . . . . . . . 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 919	. . . . . . . . . . . . . . 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 5540	. . . . . . . . . . . . . 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 12474	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 2
30		1ex 11257	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
31		vex 3484	. . . . . . . . . . . . . . . . 17 ⊢ 𝑐 ∈ V
32	30, 31	fvpr1 7212	. . . . . . . . . . . . . . . 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 3484	. . . . . . . . . . . . . . . . 17 ⊢ 𝑒 ∈ V
35	30, 34	fvpr1 7212	. . . . . . . . . . . . . . . 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 5156	. . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . 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 12343	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ V
40		vex 3484	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑑 ∈ V
41	39, 40	fvpr2 7213	. . . . . . . . . . . . . . . . 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 3484	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
44	39, 43	fvpr2 7213	. . . . . . . . . . . . . . . . 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 5156	. . . . . . . . . . . . . . 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 919	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . 16 ⊢ {1, 2} = {1, 2}
50	49, 1	prelrrx2 48634	. . . . . . . . . . . . . . 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 48634	. . . . . . . . . . . . . . 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 48641	. . . . . . . . . . . . . 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 8024	. . . . . . . . . . . . . . . 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 8024	. . . . . . . . . . . . . . . 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 5159	. . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . . 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 8025	. . . . . . . . . . . . . . . . 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 8025	. . . . . . . . . . . . . . . . 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 5159	. . . . . . . . . . . . . . 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 919	. . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}))
75		opeq2 4874	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑐 → ⟨1, 𝑥⟩ = ⟨1, 𝑐⟩)
76	75	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑑) → ⟨1, 𝑥⟩ = ⟨1, 𝑐⟩)
77		opeq2 4874	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑑 → ⟨2, 𝑦⟩ = ⟨2, 𝑑⟩)
78	77	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑑) → ⟨2, 𝑦⟩ = ⟨2, 𝑑⟩)
79	76, 78	preq12d 4741	. . . . . . . . . . . . . . . . 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 5437	. . . . . . . . . . . . . . . . 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 7585	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑐(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑑) = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
86	73, 85	sylan9eq 2797	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎) = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
87	86	eqcomd 2743	. . . . . . . . . . . . 13 ⊢ ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎))
88		fveq2 6906	. . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}))
92		opeq2 4874	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑒 → ⟨1, 𝑥⟩ = ⟨1, 𝑒⟩)
93	92	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑒 ∧ 𝑦 = 𝑓) → ⟨1, 𝑥⟩ = ⟨1, 𝑒⟩)
94		opeq2 4874	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑓 → ⟨2, 𝑦⟩ = ⟨2, 𝑓⟩)
95	94	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑒 ∧ 𝑦 = 𝑓) → ⟨2, 𝑦⟩ = ⟨2, 𝑓⟩)
96	93, 95	preq12d 4741	. . . . . . . . . . . . . . . . 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 5437	. . . . . . . . . . . . . . . . 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 7585	. . . . . . . . . . . . . . 15 ⊢ ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → (𝑒(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑓) = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
103	90, 102	sylan9eq 2797	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏) = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
104	103	eqcomd 2743	. . . . . . . . . . . . 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 5159	. . . . . . . . . . . 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 3199	. . . 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 6570	. . . 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 7338	. . . 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 7337	. . 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 848   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  Vcvv 3480  {cpr 4628  ⟨cop 4632   class class class wbr 5143  {copab 5205   × cxp 5683  –1-1-onto→wf1o 6560  ‘cfv 6561   Isom wiso 6562  (class class class)co 7431   ∈ cmpo 7433  1^st c1st 8012  2^nd c2nd 8013   ↑_m cmap 8866  ℝcr 11154  1c1 11156   < clt 11295  2c2 12321

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-2 12329

This theorem is referenced by:  rrx2plordso  48645