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Theorem rrx2plordisom 47409
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⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((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)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}
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.
StepHypRef Expression
1 rrx2plord2.r . . . . 5 𝑅 = (ℝ ↑m {1, 2})
2 eqid 2733 . . . . 5 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})
31, 2rrx2xpref1o 47404 . . . 4 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}):(ℝ × ℝ)–1-1-onto𝑅
4 elxpi 5699 . . . . . 6 (𝑎 ∈ (ℝ × ℝ) → ∃𝑐𝑑(𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)))
5 elxpi 5699 . . . . . 6 (𝑏 ∈ (ℝ × ℝ) → ∃𝑒𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)))
6 df-br 5150 . . . . . . . . . . . . 13 (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))})
7 opelxpi 5714 . . . . . . . . . . . . . . . 16 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → ⟨𝑐, 𝑑⟩ ∈ (ℝ × ℝ))
87adantl 483 . . . . . . . . . . . . . . 15 ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ⟨𝑐, 𝑑⟩ ∈ (ℝ × ℝ))
9 eleq1 2822 . . . . . . . . . . . . . . . 16 (𝑎 = ⟨𝑐, 𝑑⟩ → (𝑎 ∈ (ℝ × ℝ) ↔ ⟨𝑐, 𝑑⟩ ∈ (ℝ × ℝ)))
109adantr 482 . . . . . . . . . . . . . . 15 ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (𝑎 ∈ (ℝ × ℝ) ↔ ⟨𝑐, 𝑑⟩ ∈ (ℝ × ℝ)))
118, 10mpbird 257 . . . . . . . . . . . . . 14 ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → 𝑎 ∈ (ℝ × ℝ))
12 opelxpi 5714 . . . . . . . . . . . . . . . 16 ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → ⟨𝑒, 𝑓⟩ ∈ (ℝ × ℝ))
1312adantl 483 . . . . . . . . . . . . . . 15 ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → ⟨𝑒, 𝑓⟩ ∈ (ℝ × ℝ))
14 eleq1 2822 . . . . . . . . . . . . . . . 16 (𝑏 = ⟨𝑒, 𝑓⟩ → (𝑏 ∈ (ℝ × ℝ) ↔ ⟨𝑒, 𝑓⟩ ∈ (ℝ × ℝ)))
1514adantr 482 . . . . . . . . . . . . . . 15 ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (𝑏 ∈ (ℝ × ℝ) ↔ ⟨𝑒, 𝑓⟩ ∈ (ℝ × ℝ)))
1613, 15mpbird 257 . . . . . . . . . . . . . 14 ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → 𝑏 ∈ (ℝ × ℝ))
17 fveq2 6892 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (1st𝑥) = (1st𝑎))
18 fveq2 6892 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏 → (1st𝑦) = (1st𝑏))
1917, 18breqan12d 5165 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑎𝑦 = 𝑏) → ((1st𝑥) < (1st𝑦) ↔ (1st𝑎) < (1st𝑏)))
2017, 18eqeqan12d 2747 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑎𝑦 = 𝑏) → ((1st𝑥) = (1st𝑦) ↔ (1st𝑎) = (1st𝑏)))
21 fveq2 6892 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (2nd𝑥) = (2nd𝑎))
22 fveq2 6892 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → (2nd𝑦) = (2nd𝑏))
2321, 22breqan12d 5165 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑎𝑦 = 𝑏) → ((2nd𝑥) < (2nd𝑦) ↔ (2nd𝑎) < (2nd𝑏)))
2420, 23anbi12d 632 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑎𝑦 = 𝑏) → (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦)) ↔ ((1st𝑎) = (1st𝑏) ∧ (2nd𝑎) < (2nd𝑏))))
2519, 24orbi12d 918 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → (((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))) ↔ ((1st𝑎) < (1st𝑏) ∨ ((1st𝑎) = (1st𝑏) ∧ (2nd𝑎) < (2nd𝑏)))))
2625opelopab2a 5536 . . . . . . . . . . . . . 14 ((𝑎 ∈ (ℝ × ℝ) ∧ 𝑏 ∈ (ℝ × ℝ)) → (⟨𝑎, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))} ↔ ((1st𝑎) < (1st𝑏) ∨ ((1st𝑎) = (1st𝑏) ∧ (2nd𝑎) < (2nd𝑏)))))
2711, 16, 26syl2an 597 . . . . . . . . . . . . 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)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))} ↔ ((1st𝑎) < (1st𝑏) ∨ ((1st𝑎) = (1st𝑏) ∧ (2nd𝑎) < (2nd𝑏)))))
286, 27bitrid 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)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}𝑏 ↔ ((1st𝑎) < (1st𝑏) ∨ ((1st𝑎) = (1st𝑏) ∧ (2nd𝑎) < (2nd𝑏)))))
29 1ne2 12420 . . . . . . . . . . . . . . . 16 1 ≠ 2
30 1ex 11210 . . . . . . . . . . . . . . . . 17 1 ∈ V
31 vex 3479 . . . . . . . . . . . . . . . . 17 𝑐 ∈ V
3230, 31fvpr1 7191 . . . . . . . . . . . . . . . 16 (1 ≠ 2 → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) = 𝑐)
3329, 32mp1i 13 . . . . . . . . . . . . . . 15 (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘1) = 𝑐)
34 vex 3479 . . . . . . . . . . . . . . . . 17 𝑒 ∈ V
3530, 34fvpr1 7191 . . . . . . . . . . . . . . . 16 (1 ≠ 2 → ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) = 𝑒)
3629, 35mp1i 13 . . . . . . . . . . . . . . 15 (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘1) = 𝑒)
3733, 36breq12d 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))
3833, 36eqeq12d 2749 . . . . . . . . . . . . . . 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 12289 . . . . . . . . . . . . . . . . . 18 2 ∈ V
40 vex 3479 . . . . . . . . . . . . . . . . . 18 𝑑 ∈ V
4139, 40fvpr2 7193 . . . . . . . . . . . . . . . . 17 (1 ≠ 2 → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘2) = 𝑑)
4229, 41mp1i 13 . . . . . . . . . . . . . . . 16 (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ({⟨1, 𝑐⟩, ⟨2, 𝑑⟩}‘2) = 𝑑)
43 vex 3479 . . . . . . . . . . . . . . . . . 18 𝑓 ∈ V
4439, 43fvpr2 7193 . . . . . . . . . . . . . . . . 17 (1 ≠ 2 → ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘2) = 𝑓)
4529, 44mp1i 13 . . . . . . . . . . . . . . . 16 (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ({⟨1, 𝑒⟩, ⟨2, 𝑓⟩}‘2) = 𝑓)
4642, 45breq12d 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))
4738, 46anbi12d 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 = 𝑒𝑑 < 𝑓)))
4837, 47orbi12d 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 = 𝑒𝑑 < 𝑓))))
49 eqid 2733 . . . . . . . . . . . . . . . 16 {1, 2} = {1, 2}
5049, 1prelrrx2 47399 . . . . . . . . . . . . . . 15 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} ∈ 𝑅)
5150adantl 483 . . . . . . . . . . . . . 14 ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} ∈ 𝑅)
5249, 1prelrrx2 47399 . . . . . . . . . . . . . . 15 ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → {⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ∈ 𝑅)
5352adantl 483 . . . . . . . . . . . . . 14 ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → {⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ∈ 𝑅)
54 rrx2plord.o . . . . . . . . . . . . . . 15 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}
5554rrx2plord 47406 . . . . . . . . . . . . . 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)))))
5651, 53, 55syl2an 597 . . . . . . . . . . . . 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)))))
5731, 40op1std 7985 . . . . . . . . . . . . . . . 16 (𝑎 = ⟨𝑐, 𝑑⟩ → (1st𝑎) = 𝑐)
5857adantr 482 . . . . . . . . . . . . . . 15 ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (1st𝑎) = 𝑐)
5934, 43op1std 7985 . . . . . . . . . . . . . . . 16 (𝑏 = ⟨𝑒, 𝑓⟩ → (1st𝑏) = 𝑒)
6059adantr 482 . . . . . . . . . . . . . . 15 ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (1st𝑏) = 𝑒)
6158, 60breqan12d 5165 . . . . . . . . . . . . . 14 (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ((1st𝑎) < (1st𝑏) ↔ 𝑐 < 𝑒))
6258, 60eqeqan12d 2747 . . . . . . . . . . . . . . 15 (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ((1st𝑎) = (1st𝑏) ↔ 𝑐 = 𝑒))
6331, 40op2ndd 7986 . . . . . . . . . . . . . . . . 17 (𝑎 = ⟨𝑐, 𝑑⟩ → (2nd𝑎) = 𝑑)
6463adantr 482 . . . . . . . . . . . . . . . 16 ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (2nd𝑎) = 𝑑)
6534, 43op2ndd 7986 . . . . . . . . . . . . . . . . 17 (𝑏 = ⟨𝑒, 𝑓⟩ → (2nd𝑏) = 𝑓)
6665adantr 482 . . . . . . . . . . . . . . . 16 ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (2nd𝑏) = 𝑓)
6764, 66breqan12d 5165 . . . . . . . . . . . . . . 15 (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → ((2nd𝑎) < (2nd𝑏) ↔ 𝑑 < 𝑓))
6862, 67anbi12d 632 . . . . . . . . . . . . . 14 (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (((1st𝑎) = (1st𝑏) ∧ (2nd𝑎) < (2nd𝑏)) ↔ (𝑐 = 𝑒𝑑 < 𝑓)))
6961, 68orbi12d 918 . . . . . . . . . . . . 13 (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (((1st𝑎) < (1st𝑏) ∨ ((1st𝑎) = (1st𝑏) ∧ (2nd𝑎) < (2nd𝑏))) ↔ (𝑐 < 𝑒 ∨ (𝑐 = 𝑒𝑑 < 𝑓))))
7048, 56, 693bitr4rd 312 . . . . . . . . . . . 12 (((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ (𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (((1st𝑎) < (1st𝑏) ∨ ((1st𝑎) = (1st𝑏) ∧ (2nd𝑎) < (2nd𝑏))) ↔ {⟨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 7412 . . . . . . . . . . . . . . . 16 (𝑐(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑑) = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘⟨𝑐, 𝑑⟩)
7371, 72eqtr4di 2791 . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . . 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⟩)
7675adantr 482 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑐𝑦 = 𝑑) → ⟨1, 𝑥⟩ = ⟨1, 𝑐⟩)
77 opeq2 4875 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑑 → ⟨2, 𝑦⟩ = ⟨2, 𝑑⟩)
7877adantl 483 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑐𝑦 = 𝑑) → ⟨2, 𝑦⟩ = ⟨2, 𝑑⟩)
7976, 78preq12d 4746 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑐𝑦 = 𝑑) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
8079adantl 483 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) ∧ (𝑥 = 𝑐𝑦 = 𝑑)) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
81 simpl 484 . . . . . . . . . . . . . . . 16 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → 𝑐 ∈ ℝ)
82 simpr 486 . . . . . . . . . . . . . . . 16 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → 𝑑 ∈ ℝ)
83 prex 5433 . . . . . . . . . . . . . . . . 17 {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} ∈ V
8483a1i 11 . . . . . . . . . . . . . . . 16 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → {⟨1, 𝑐⟩, ⟨2, 𝑑⟩} ∈ V)
8574, 80, 81, 82, 84ovmpod 7560 . . . . . . . . . . . . . . 15 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑐(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑑) = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
8673, 85sylan9eq 2793 . . . . . . . . . . . . . 14 ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎) = {⟨1, 𝑐⟩, ⟨2, 𝑑⟩})
8786eqcomd 2739 . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . . . 16 (𝑒(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑓) = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘⟨𝑒, 𝑓⟩)
9088, 89eqtr4di 2791 . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . . 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⟩)
9392adantr 482 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑒𝑦 = 𝑓) → ⟨1, 𝑥⟩ = ⟨1, 𝑒⟩)
94 opeq2 4875 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑓 → ⟨2, 𝑦⟩ = ⟨2, 𝑓⟩)
9594adantl 483 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑒𝑦 = 𝑓) → ⟨2, 𝑦⟩ = ⟨2, 𝑓⟩)
9693, 95preq12d 4746 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑒𝑦 = 𝑓) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
9796adantl 483 . . . . . . . . . . . . . . . 16 (((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) ∧ (𝑥 = 𝑒𝑦 = 𝑓)) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
98 simpl 484 . . . . . . . . . . . . . . . 16 ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → 𝑒 ∈ ℝ)
99 simpr 486 . . . . . . . . . . . . . . . 16 ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → 𝑓 ∈ ℝ)
100 prex 5433 . . . . . . . . . . . . . . . . 17 {⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ∈ V
101100a1i 11 . . . . . . . . . . . . . . . 16 ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → {⟨1, 𝑒⟩, ⟨2, 𝑓⟩} ∈ V)
10291, 97, 98, 99, 101ovmpod 7560 . . . . . . . . . . . . . . 15 ((𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ) → (𝑒(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})𝑓) = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
10390, 102sylan9eq 2793 . . . . . . . . . . . . . 14 ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏) = {⟨1, 𝑒⟩, ⟨2, 𝑓⟩})
104103eqcomd 2739 . . . . . . . . . . . . 13 ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → {⟨1, 𝑒⟩, ⟨2, 𝑓⟩} = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))
10587, 104breqan12d 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)))
10628, 70, 1053bitrd 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)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏)))
107106expcom 415 . . . . . . . . . 10 ((𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))))
108107exlimivv 1936 . . . . . . . . 9 (∃𝑒𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))))
109108com12 32 . . . . . . . 8 ((𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (∃𝑒𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))))
110109exlimivv 1936 . . . . . . 7 (∃𝑐𝑑(𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (∃𝑒𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ)) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))))
111110imp 408 . . . . . 6 ((∃𝑐𝑑(𝑎 = ⟨𝑐, 𝑑⟩ ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) ∧ ∃𝑒𝑓(𝑏 = ⟨𝑒, 𝑓⟩ ∧ (𝑒 ∈ ℝ ∧ 𝑓 ∈ ℝ))) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏)))
1124, 5, 111syl2an 597 . . . . 5 ((𝑎 ∈ (ℝ × ℝ) ∧ 𝑏 ∈ (ℝ × ℝ)) → (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏)))
113112rgen2 3198 . . . 4 𝑎 ∈ (ℝ × ℝ)∀𝑏 ∈ (ℝ × ℝ)(𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}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)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}, 𝑂((ℝ × ℝ), 𝑅) ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}):(ℝ × ℝ)–1-1-onto𝑅 ∧ ∀𝑎 ∈ (ℝ × ℝ)∀𝑏 ∈ (ℝ × ℝ)(𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}𝑏 ↔ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑎)𝑂((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})‘𝑏))))
1153, 113, 114mpbir2an 710 . . 3 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}, 𝑂((ℝ × ℝ), 𝑅)
116 rrx2plordisom.t . . . 4 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}
117 isoeq2 7315 . . . 4 (𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))} → ((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)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}, 𝑂((ℝ × ℝ), 𝑅)))
118116, 117ax-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)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}, 𝑂((ℝ × ℝ), 𝑅))
119115, 118mpbir 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 7314 . . 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)))
122120, 121ax-mp 5 . 2 (𝐹 Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅) ↔ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅))
123119, 122mpbir 230 1 𝐹 Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wex 1782  wcel 2107  wne 2941  wral 3062  Vcvv 3475  {cpr 4631  cop 4635   class class class wbr 5149  {copab 5211   × cxp 5675  1-1-ontowf1o 6543  cfv 6544   Isom wiso 6545  (class class class)co 7409  cmpo 7411  1st c1st 7973  2nd c2nd 7974  m cmap 8820  cr 11109  1c1 11111   < clt 11248  2c2 12267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-2 12275
This theorem is referenced by:  rrx2plordso  47410
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