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Theorem brafs 31107
Description: Binary relation form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-2013.)
Hypotheses
Ref Expression
brafs.p 𝑃 = (Base‘𝐺)
brafs.d = (dist‘𝐺)
brafs.i 𝐼 = (Itv‘𝐺)
brafs.g (𝜑𝐺 ∈ TarskiG)
brafs.o 𝑂 = (AFS‘𝐺)
brafs.1 (𝜑𝐴𝑃)
brafs.2 (𝜑𝐵𝑃)
brafs.3 (𝜑𝐶𝑃)
brafs.4 (𝜑𝐷𝑃)
brafs.5 (𝜑𝑋𝑃)
brafs.6 (𝜑𝑌𝑃)
brafs.7 (𝜑𝑍𝑃)
brafs.8 (𝜑𝑊𝑃)
Assertion
Ref Expression
brafs (𝜑 → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑂⟨⟨𝑋, 𝑌⟩, ⟨𝑍, 𝑊⟩⟩ ↔ ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ ((𝐴 𝐵) = (𝑋 𝑌) ∧ (𝐵 𝐶) = (𝑌 𝑍)) ∧ ((𝐴 𝐷) = (𝑋 𝑊) ∧ (𝐵 𝐷) = (𝑌 𝑊)))))

Proof of Theorem brafs
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6819 . . . . 5 (𝑎 = 𝐴 → (𝑎𝐼𝑐) = (𝐴𝐼𝑐))
21eleq2d 2839 . . . 4 (𝑎 = 𝐴 → (𝑏 ∈ (𝑎𝐼𝑐) ↔ 𝑏 ∈ (𝐴𝐼𝑐)))
32anbi1d 616 . . 3 (𝑎 = 𝐴 → ((𝑏 ∈ (𝑎𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑏 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧))))
4 oveq1 6819 . . . . 5 (𝑎 = 𝐴 → (𝑎 𝑏) = (𝐴 𝑏))
54eqeq1d 2776 . . . 4 (𝑎 = 𝐴 → ((𝑎 𝑏) = (𝑥 𝑦) ↔ (𝐴 𝑏) = (𝑥 𝑦)))
65anbi1d 616 . . 3 (𝑎 = 𝐴 → (((𝑎 𝑏) = (𝑥 𝑦) ∧ (𝑏 𝑐) = (𝑦 𝑧)) ↔ ((𝐴 𝑏) = (𝑥 𝑦) ∧ (𝑏 𝑐) = (𝑦 𝑧))))
7 oveq1 6819 . . . . 5 (𝑎 = 𝐴 → (𝑎 𝑑) = (𝐴 𝑑))
87eqeq1d 2776 . . . 4 (𝑎 = 𝐴 → ((𝑎 𝑑) = (𝑥 𝑤) ↔ (𝐴 𝑑) = (𝑥 𝑤)))
98anbi1d 616 . . 3 (𝑎 = 𝐴 → (((𝑎 𝑑) = (𝑥 𝑤) ∧ (𝑏 𝑑) = (𝑦 𝑤)) ↔ ((𝐴 𝑑) = (𝑥 𝑤) ∧ (𝑏 𝑑) = (𝑦 𝑤))))
103, 6, 93anbi123d 1550 . 2 (𝑎 = 𝐴 → (((𝑏 ∈ (𝑎𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝑎 𝑏) = (𝑥 𝑦) ∧ (𝑏 𝑐) = (𝑦 𝑧)) ∧ ((𝑎 𝑑) = (𝑥 𝑤) ∧ (𝑏 𝑑) = (𝑦 𝑤))) ↔ ((𝑏 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝐴 𝑏) = (𝑥 𝑦) ∧ (𝑏 𝑐) = (𝑦 𝑧)) ∧ ((𝐴 𝑑) = (𝑥 𝑤) ∧ (𝑏 𝑑) = (𝑦 𝑤)))))
11 eleq1 2841 . . . 4 (𝑏 = 𝐵 → (𝑏 ∈ (𝐴𝐼𝑐) ↔ 𝐵 ∈ (𝐴𝐼𝑐)))
1211anbi1d 616 . . 3 (𝑏 = 𝐵 → ((𝑏 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧))))
13 oveq2 6820 . . . . 5 (𝑏 = 𝐵 → (𝐴 𝑏) = (𝐴 𝐵))
1413eqeq1d 2776 . . . 4 (𝑏 = 𝐵 → ((𝐴 𝑏) = (𝑥 𝑦) ↔ (𝐴 𝐵) = (𝑥 𝑦)))
15 oveq1 6819 . . . . 5 (𝑏 = 𝐵 → (𝑏 𝑐) = (𝐵 𝑐))
1615eqeq1d 2776 . . . 4 (𝑏 = 𝐵 → ((𝑏 𝑐) = (𝑦 𝑧) ↔ (𝐵 𝑐) = (𝑦 𝑧)))
1714, 16anbi12d 617 . . 3 (𝑏 = 𝐵 → (((𝐴 𝑏) = (𝑥 𝑦) ∧ (𝑏 𝑐) = (𝑦 𝑧)) ↔ ((𝐴 𝐵) = (𝑥 𝑦) ∧ (𝐵 𝑐) = (𝑦 𝑧))))
18 oveq1 6819 . . . . 5 (𝑏 = 𝐵 → (𝑏 𝑑) = (𝐵 𝑑))
1918eqeq1d 2776 . . . 4 (𝑏 = 𝐵 → ((𝑏 𝑑) = (𝑦 𝑤) ↔ (𝐵 𝑑) = (𝑦 𝑤)))
2019anbi2d 615 . . 3 (𝑏 = 𝐵 → (((𝐴 𝑑) = (𝑥 𝑤) ∧ (𝑏 𝑑) = (𝑦 𝑤)) ↔ ((𝐴 𝑑) = (𝑥 𝑤) ∧ (𝐵 𝑑) = (𝑦 𝑤))))
2112, 17, 203anbi123d 1550 . 2 (𝑏 = 𝐵 → (((𝑏 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝐴 𝑏) = (𝑥 𝑦) ∧ (𝑏 𝑐) = (𝑦 𝑧)) ∧ ((𝐴 𝑑) = (𝑥 𝑤) ∧ (𝑏 𝑑) = (𝑦 𝑤))) ↔ ((𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝐴 𝐵) = (𝑥 𝑦) ∧ (𝐵 𝑐) = (𝑦 𝑧)) ∧ ((𝐴 𝑑) = (𝑥 𝑤) ∧ (𝐵 𝑑) = (𝑦 𝑤)))))
22 oveq2 6820 . . . . 5 (𝑐 = 𝐶 → (𝐴𝐼𝑐) = (𝐴𝐼𝐶))
2322eleq2d 2839 . . . 4 (𝑐 = 𝐶 → (𝐵 ∈ (𝐴𝐼𝑐) ↔ 𝐵 ∈ (𝐴𝐼𝐶)))
2423anbi1d 616 . . 3 (𝑐 = 𝐶 → ((𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑦 ∈ (𝑥𝐼𝑧))))
25 oveq2 6820 . . . . 5 (𝑐 = 𝐶 → (𝐵 𝑐) = (𝐵 𝐶))
2625eqeq1d 2776 . . . 4 (𝑐 = 𝐶 → ((𝐵 𝑐) = (𝑦 𝑧) ↔ (𝐵 𝐶) = (𝑦 𝑧)))
2726anbi2d 615 . . 3 (𝑐 = 𝐶 → (((𝐴 𝐵) = (𝑥 𝑦) ∧ (𝐵 𝑐) = (𝑦 𝑧)) ↔ ((𝐴 𝐵) = (𝑥 𝑦) ∧ (𝐵 𝐶) = (𝑦 𝑧))))
2824, 273anbi12d 1551 . 2 (𝑐 = 𝐶 → (((𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝐴 𝐵) = (𝑥 𝑦) ∧ (𝐵 𝑐) = (𝑦 𝑧)) ∧ ((𝐴 𝑑) = (𝑥 𝑤) ∧ (𝐵 𝑑) = (𝑦 𝑤))) ↔ ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝐴 𝐵) = (𝑥 𝑦) ∧ (𝐵 𝐶) = (𝑦 𝑧)) ∧ ((𝐴 𝑑) = (𝑥 𝑤) ∧ (𝐵 𝑑) = (𝑦 𝑤)))))
29 oveq2 6820 . . . . 5 (𝑑 = 𝐷 → (𝐴 𝑑) = (𝐴 𝐷))
3029eqeq1d 2776 . . . 4 (𝑑 = 𝐷 → ((𝐴 𝑑) = (𝑥 𝑤) ↔ (𝐴 𝐷) = (𝑥 𝑤)))
31 oveq2 6820 . . . . 5 (𝑑 = 𝐷 → (𝐵 𝑑) = (𝐵 𝐷))
3231eqeq1d 2776 . . . 4 (𝑑 = 𝐷 → ((𝐵 𝑑) = (𝑦 𝑤) ↔ (𝐵 𝐷) = (𝑦 𝑤)))
3330, 32anbi12d 617 . . 3 (𝑑 = 𝐷 → (((𝐴 𝑑) = (𝑥 𝑤) ∧ (𝐵 𝑑) = (𝑦 𝑤)) ↔ ((𝐴 𝐷) = (𝑥 𝑤) ∧ (𝐵 𝐷) = (𝑦 𝑤))))
34333anbi3d 1556 . 2 (𝑑 = 𝐷 → (((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝐴 𝐵) = (𝑥 𝑦) ∧ (𝐵 𝐶) = (𝑦 𝑧)) ∧ ((𝐴 𝑑) = (𝑥 𝑤) ∧ (𝐵 𝑑) = (𝑦 𝑤))) ↔ ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝐴 𝐵) = (𝑥 𝑦) ∧ (𝐵 𝐶) = (𝑦 𝑧)) ∧ ((𝐴 𝐷) = (𝑥 𝑤) ∧ (𝐵 𝐷) = (𝑦 𝑤)))))
35 oveq1 6819 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
3635eleq2d 2839 . . . 4 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
3736anbi2d 615 . . 3 (𝑥 = 𝑋 → ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑦 ∈ (𝑋𝐼𝑧))))
38 oveq1 6819 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
3938eqeq2d 2784 . . . 4 (𝑥 = 𝑋 → ((𝐴 𝐵) = (𝑥 𝑦) ↔ (𝐴 𝐵) = (𝑋 𝑦)))
4039anbi1d 616 . . 3 (𝑥 = 𝑋 → (((𝐴 𝐵) = (𝑥 𝑦) ∧ (𝐵 𝐶) = (𝑦 𝑧)) ↔ ((𝐴 𝐵) = (𝑋 𝑦) ∧ (𝐵 𝐶) = (𝑦 𝑧))))
41 oveq1 6819 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑤) = (𝑋 𝑤))
4241eqeq2d 2784 . . . 4 (𝑥 = 𝑋 → ((𝐴 𝐷) = (𝑥 𝑤) ↔ (𝐴 𝐷) = (𝑋 𝑤)))
4342anbi1d 616 . . 3 (𝑥 = 𝑋 → (((𝐴 𝐷) = (𝑥 𝑤) ∧ (𝐵 𝐷) = (𝑦 𝑤)) ↔ ((𝐴 𝐷) = (𝑋 𝑤) ∧ (𝐵 𝐷) = (𝑦 𝑤))))
4437, 40, 433anbi123d 1550 . 2 (𝑥 = 𝑋 → (((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝐴 𝐵) = (𝑥 𝑦) ∧ (𝐵 𝐶) = (𝑦 𝑧)) ∧ ((𝐴 𝐷) = (𝑥 𝑤) ∧ (𝐵 𝐷) = (𝑦 𝑤))) ↔ ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑦 ∈ (𝑋𝐼𝑧)) ∧ ((𝐴 𝐵) = (𝑋 𝑦) ∧ (𝐵 𝐶) = (𝑦 𝑧)) ∧ ((𝐴 𝐷) = (𝑋 𝑤) ∧ (𝐵 𝐷) = (𝑦 𝑤)))))
45 eleq1 2841 . . . 4 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
4645anbi2d 615 . . 3 (𝑦 = 𝑌 → ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑦 ∈ (𝑋𝐼𝑧)) ↔ (𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑌 ∈ (𝑋𝐼𝑧))))
47 oveq2 6820 . . . . 5 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
4847eqeq2d 2784 . . . 4 (𝑦 = 𝑌 → ((𝐴 𝐵) = (𝑋 𝑦) ↔ (𝐴 𝐵) = (𝑋 𝑌)))
49 oveq1 6819 . . . . 5 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
5049eqeq2d 2784 . . . 4 (𝑦 = 𝑌 → ((𝐵 𝐶) = (𝑦 𝑧) ↔ (𝐵 𝐶) = (𝑌 𝑧)))
5148, 50anbi12d 617 . . 3 (𝑦 = 𝑌 → (((𝐴 𝐵) = (𝑋 𝑦) ∧ (𝐵 𝐶) = (𝑦 𝑧)) ↔ ((𝐴 𝐵) = (𝑋 𝑌) ∧ (𝐵 𝐶) = (𝑌 𝑧))))
52 oveq1 6819 . . . . 5 (𝑦 = 𝑌 → (𝑦 𝑤) = (𝑌 𝑤))
5352eqeq2d 2784 . . . 4 (𝑦 = 𝑌 → ((𝐵 𝐷) = (𝑦 𝑤) ↔ (𝐵 𝐷) = (𝑌 𝑤)))
5453anbi2d 615 . . 3 (𝑦 = 𝑌 → (((𝐴 𝐷) = (𝑋 𝑤) ∧ (𝐵 𝐷) = (𝑦 𝑤)) ↔ ((𝐴 𝐷) = (𝑋 𝑤) ∧ (𝐵 𝐷) = (𝑌 𝑤))))
5546, 51, 543anbi123d 1550 . 2 (𝑦 = 𝑌 → (((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑦 ∈ (𝑋𝐼𝑧)) ∧ ((𝐴 𝐵) = (𝑋 𝑦) ∧ (𝐵 𝐶) = (𝑦 𝑧)) ∧ ((𝐴 𝐷) = (𝑋 𝑤) ∧ (𝐵 𝐷) = (𝑦 𝑤))) ↔ ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑌 ∈ (𝑋𝐼𝑧)) ∧ ((𝐴 𝐵) = (𝑋 𝑌) ∧ (𝐵 𝐶) = (𝑌 𝑧)) ∧ ((𝐴 𝐷) = (𝑋 𝑤) ∧ (𝐵 𝐷) = (𝑌 𝑤)))))
56 oveq2 6820 . . . . 5 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
5756eleq2d 2839 . . . 4 (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
5857anbi2d 615 . . 3 (𝑧 = 𝑍 → ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑌 ∈ (𝑋𝐼𝑧)) ↔ (𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑌 ∈ (𝑋𝐼𝑍))))
59 oveq2 6820 . . . . 5 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
6059eqeq2d 2784 . . . 4 (𝑧 = 𝑍 → ((𝐵 𝐶) = (𝑌 𝑧) ↔ (𝐵 𝐶) = (𝑌 𝑍)))
6160anbi2d 615 . . 3 (𝑧 = 𝑍 → (((𝐴 𝐵) = (𝑋 𝑌) ∧ (𝐵 𝐶) = (𝑌 𝑧)) ↔ ((𝐴 𝐵) = (𝑋 𝑌) ∧ (𝐵 𝐶) = (𝑌 𝑍))))
6258, 613anbi12d 1551 . 2 (𝑧 = 𝑍 → (((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑌 ∈ (𝑋𝐼𝑧)) ∧ ((𝐴 𝐵) = (𝑋 𝑌) ∧ (𝐵 𝐶) = (𝑌 𝑧)) ∧ ((𝐴 𝐷) = (𝑋 𝑤) ∧ (𝐵 𝐷) = (𝑌 𝑤))) ↔ ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ ((𝐴 𝐵) = (𝑋 𝑌) ∧ (𝐵 𝐶) = (𝑌 𝑍)) ∧ ((𝐴 𝐷) = (𝑋 𝑤) ∧ (𝐵 𝐷) = (𝑌 𝑤)))))
63 oveq2 6820 . . . . 5 (𝑤 = 𝑊 → (𝑋 𝑤) = (𝑋 𝑊))
6463eqeq2d 2784 . . . 4 (𝑤 = 𝑊 → ((𝐴 𝐷) = (𝑋 𝑤) ↔ (𝐴 𝐷) = (𝑋 𝑊)))
65 oveq2 6820 . . . . 5 (𝑤 = 𝑊 → (𝑌 𝑤) = (𝑌 𝑊))
6665eqeq2d 2784 . . . 4 (𝑤 = 𝑊 → ((𝐵 𝐷) = (𝑌 𝑤) ↔ (𝐵 𝐷) = (𝑌 𝑊)))
6764, 66anbi12d 617 . . 3 (𝑤 = 𝑊 → (((𝐴 𝐷) = (𝑋 𝑤) ∧ (𝐵 𝐷) = (𝑌 𝑤)) ↔ ((𝐴 𝐷) = (𝑋 𝑊) ∧ (𝐵 𝐷) = (𝑌 𝑊))))
68673anbi3d 1556 . 2 (𝑤 = 𝑊 → (((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ ((𝐴 𝐵) = (𝑋 𝑌) ∧ (𝐵 𝐶) = (𝑌 𝑍)) ∧ ((𝐴 𝐷) = (𝑋 𝑤) ∧ (𝐵 𝐷) = (𝑌 𝑤))) ↔ ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ ((𝐴 𝐵) = (𝑋 𝑌) ∧ (𝐵 𝐶) = (𝑌 𝑍)) ∧ ((𝐴 𝐷) = (𝑋 𝑊) ∧ (𝐵 𝐷) = (𝑌 𝑊)))))
69 brafs.o . . 3 𝑂 = (AFS‘𝐺)
70 brafs.p . . . 4 𝑃 = (Base‘𝐺)
71 brafs.d . . . 4 = (dist‘𝐺)
72 brafs.i . . . 4 𝐼 = (Itv‘𝐺)
73 brafs.g . . . 4 (𝜑𝐺 ∈ TarskiG)
7470, 71, 72, 73afsval 31106 . . 3 (𝜑 → (AFS‘𝐺) = {⟨𝑒, 𝑓⟩ ∣ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃𝑥𝑃𝑦𝑃𝑧𝑃𝑤𝑃 (𝑒 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑓 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ ((𝑏 ∈ (𝑎𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝑎 𝑏) = (𝑥 𝑦) ∧ (𝑏 𝑐) = (𝑦 𝑧)) ∧ ((𝑎 𝑑) = (𝑥 𝑤) ∧ (𝑏 𝑑) = (𝑦 𝑤))))})
7569, 74syl5eq 2820 . 2 (𝜑𝑂 = {⟨𝑒, 𝑓⟩ ∣ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃𝑥𝑃𝑦𝑃𝑧𝑃𝑤𝑃 (𝑒 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑓 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ ((𝑏 ∈ (𝑎𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝑎 𝑏) = (𝑥 𝑦) ∧ (𝑏 𝑐) = (𝑦 𝑧)) ∧ ((𝑎 𝑑) = (𝑥 𝑤) ∧ (𝑏 𝑑) = (𝑦 𝑤))))})
76 brafs.1 . 2 (𝜑𝐴𝑃)
77 brafs.2 . 2 (𝜑𝐵𝑃)
78 brafs.3 . 2 (𝜑𝐶𝑃)
79 brafs.4 . 2 (𝜑𝐷𝑃)
80 brafs.5 . 2 (𝜑𝑋𝑃)
81 brafs.6 . 2 (𝜑𝑌𝑃)
82 brafs.7 . 2 (𝜑𝑍𝑃)
83 brafs.8 . 2 (𝜑𝑊𝑃)
8410, 21, 28, 34, 44, 55, 62, 68, 75, 76, 77, 78, 79, 80, 81, 82, 83br8d 29777 1 (𝜑 → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑂⟨⟨𝑋, 𝑌⟩, ⟨𝑍, 𝑊⟩⟩ ↔ ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ ((𝐴 𝐵) = (𝑋 𝑌) ∧ (𝐵 𝐶) = (𝑌 𝑍)) ∧ ((𝐴 𝐷) = (𝑋 𝑊) ∧ (𝐵 𝐷) = (𝑌 𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 383  w3a 1098   = wceq 1634  wcel 2148  wrex 3065  cop 4332   class class class wbr 4797  {copab 4859  cfv 6042  (class class class)co 6812  Basecbs 16084  distcds 16178  TarskiGcstrkg 25571  Itvcitv 25577  AFScafs 31104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-iota 6005  df-fun 6044  df-fv 6050  df-ov 6815  df-afs 31105
This theorem is referenced by:  tg5segofs  31108
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