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Theorem frpoins3xpg 8164
Description: Special case of founded partial induction over a Cartesian product. (Contributed by Scott Fenton, 22-Aug-2024.)
Hypotheses
Ref Expression
frpoins3xpg.1 ((𝑥𝐴𝑦𝐵) → (∀𝑧𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒) → 𝜑))
frpoins3xpg.2 (𝑥 = 𝑧 → (𝜑𝜓))
frpoins3xpg.3 (𝑦 = 𝑤 → (𝜓𝜒))
frpoins3xpg.4 (𝑥 = 𝑋 → (𝜑𝜃))
frpoins3xpg.5 (𝑦 = 𝑌 → (𝜃𝜏))
Assertion
Ref Expression
frpoins3xpg (((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) ∧ (𝑋𝐴𝑌𝐵)) → 𝜏)
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧,𝑤   𝑤,𝐵,𝑥,𝑦,𝑧   𝜒,𝑦   𝜑,𝑧   𝜓,𝑥,𝑤   𝑤,𝑅,𝑥,𝑦,𝑧   𝜏,𝑦   𝜃,𝑥   𝑥,𝑋,𝑦   𝑦,𝑌
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑤)   𝜓(𝑦,𝑧)   𝜒(𝑥,𝑧,𝑤)   𝜃(𝑦,𝑧,𝑤)   𝜏(𝑥,𝑧,𝑤)   𝑋(𝑧,𝑤)   𝑌(𝑥,𝑧,𝑤)

Proof of Theorem frpoins3xpg
Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp2 5713 . . . . . 6 (𝑝 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑝 = ⟨𝑥, 𝑦⟩)
2 nfcv 2903 . . . . . . . . 9 𝑥Pred(𝑅, (𝐴 × 𝐵), 𝑝)
3 nfsbc1v 3811 . . . . . . . . 9 𝑥[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑
42, 3nfralw 3309 . . . . . . . 8 𝑥𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑
5 nfsbc1v 3811 . . . . . . . 8 𝑥[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑
64, 5nfim 1894 . . . . . . 7 𝑥(∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑)
7 nfv 1912 . . . . . . . 8 𝑦 𝑥𝐴
8 nfcv 2903 . . . . . . . . . 10 𝑦Pred(𝑅, (𝐴 × 𝐵), 𝑝)
9 nfcv 2903 . . . . . . . . . . 11 𝑦(1st𝑞)
10 nfsbc1v 3811 . . . . . . . . . . 11 𝑦[(2nd𝑞) / 𝑦]𝜑
119, 10nfsbcw 3813 . . . . . . . . . 10 𝑦[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑
128, 11nfralw 3309 . . . . . . . . 9 𝑦𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑
13 nfcv 2903 . . . . . . . . . 10 𝑦(1st𝑝)
14 nfsbc1v 3811 . . . . . . . . . 10 𝑦[(2nd𝑝) / 𝑦]𝜑
1513, 14nfsbcw 3813 . . . . . . . . 9 𝑦[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑
1612, 15nfim 1894 . . . . . . . 8 𝑦(∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑)
17 frpoins3xpg.2 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝜑𝜓))
1817sbcbidv 3851 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ([(2nd𝑞) / 𝑦]𝜑[(2nd𝑞) / 𝑦]𝜓))
1918cbvsbcvw 3826 . . . . . . . . . . . . . 14 ([(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑦]𝜓)
20 frpoins3xpg.3 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑤 → (𝜓𝜒))
2120cbvsbcvw 3826 . . . . . . . . . . . . . . 15 ([(2nd𝑞) / 𝑦]𝜓[(2nd𝑞) / 𝑤]𝜒)
2221sbcbii 3852 . . . . . . . . . . . . . 14 ([(1st𝑞) / 𝑧][(2nd𝑞) / 𝑦]𝜓[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)
2319, 22bitri 275 . . . . . . . . . . . . 13 ([(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)
2423ralbii 3091 . . . . . . . . . . . 12 (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑 ↔ ∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)
25 impexp 450 . . . . . . . . . . . . . . 15 (((𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒) ↔ (𝑞 ∈ (𝐴 × 𝐵) → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)))
26 elin 3979 . . . . . . . . . . . . . . . . 17 (𝑞 ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
27 predss 6331 . . . . . . . . . . . . . . . . . . 19 Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) ⊆ (𝐴 × 𝐵)
28 sseqin2 4231 . . . . . . . . . . . . . . . . . . 19 (Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) ⊆ (𝐴 × 𝐵) ↔ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) = Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
2927, 28mpbi 230 . . . . . . . . . . . . . . . . . 18 ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) = Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
3029eleq2i 2831 . . . . . . . . . . . . . . . . 17 (𝑞 ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
3126, 30bitr3i 277 . . . . . . . . . . . . . . . 16 ((𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
3231imbi1i 349 . . . . . . . . . . . . . . 15 (((𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒) ↔ (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒))
3325, 32bitr3i 277 . . . . . . . . . . . . . 14 ((𝑞 ∈ (𝐴 × 𝐵) → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)) ↔ (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒))
3433albii 1816 . . . . . . . . . . . . 13 (∀𝑞(𝑞 ∈ (𝐴 × 𝐵) → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)) ↔ ∀𝑞(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒))
35 df-ral 3060 . . . . . . . . . . . . 13 (∀𝑞 ∈ (𝐴 × 𝐵)(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒) ↔ ∀𝑞(𝑞 ∈ (𝐴 × 𝐵) → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)))
36 df-ral 3060 . . . . . . . . . . . . 13 (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒 ↔ ∀𝑞(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒))
3734, 35, 363bitr4ri 304 . . . . . . . . . . . 12 (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒 ↔ ∀𝑞 ∈ (𝐴 × 𝐵)(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒))
38 nfv 1912 . . . . . . . . . . . . . . 15 𝑧 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
39 nfsbc1v 3811 . . . . . . . . . . . . . . 15 𝑧[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒
4038, 39nfim 1894 . . . . . . . . . . . . . 14 𝑧(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)
41 nfv 1912 . . . . . . . . . . . . . . 15 𝑤 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
42 nfcv 2903 . . . . . . . . . . . . . . . 16 𝑤(1st𝑞)
43 nfsbc1v 3811 . . . . . . . . . . . . . . . 16 𝑤[(2nd𝑞) / 𝑤]𝜒
4442, 43nfsbcw 3813 . . . . . . . . . . . . . . 15 𝑤[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒
4541, 44nfim 1894 . . . . . . . . . . . . . 14 𝑤(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)
46 nfv 1912 . . . . . . . . . . . . . 14 𝑞(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)
47 eleq1 2827 . . . . . . . . . . . . . . 15 (𝑞 = ⟨𝑧, 𝑤⟩ → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
48 sbcopeq1a 8073 . . . . . . . . . . . . . . 15 (𝑞 = ⟨𝑧, 𝑤⟩ → ([(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒𝜒))
4947, 48imbi12d 344 . . . . . . . . . . . . . 14 (𝑞 = ⟨𝑧, 𝑤⟩ → ((𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒) ↔ (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)))
5040, 45, 46, 49ralxpf 5860 . . . . . . . . . . . . 13 (∀𝑞 ∈ (𝐴 × 𝐵)(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒) ↔ ∀𝑧𝐴𝑤𝐵 (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
51 r2al 3193 . . . . . . . . . . . . 13 (∀𝑧𝐴𝑤𝐵 (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒) ↔ ∀𝑧𝑤((𝑧𝐴𝑤𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)))
52 impexp 450 . . . . . . . . . . . . . . . 16 (((⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) → 𝜒) ↔ (⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)))
53 opelxp 5725 . . . . . . . . . . . . . . . . 17 (⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ↔ (𝑧𝐴𝑤𝐵))
5453imbi1i 349 . . . . . . . . . . . . . . . 16 ((⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)) ↔ ((𝑧𝐴𝑤𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)))
5552, 54bitri 275 . . . . . . . . . . . . . . 15 (((⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) → 𝜒) ↔ ((𝑧𝐴𝑤𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)))
56 elin 3979 . . . . . . . . . . . . . . . . 17 (⟨𝑧, 𝑤⟩ ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ (⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
5729eleq2i 2831 . . . . . . . . . . . . . . . . 17 (⟨𝑧, 𝑤⟩ ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
5856, 57bitr3i 277 . . . . . . . . . . . . . . . 16 ((⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
5958imbi1i 349 . . . . . . . . . . . . . . 15 (((⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) → 𝜒) ↔ (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
6055, 59bitr3i 277 . . . . . . . . . . . . . 14 (((𝑧𝐴𝑤𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)) ↔ (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
61602albii 1817 . . . . . . . . . . . . 13 (∀𝑧𝑤((𝑧𝐴𝑤𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)) ↔ ∀𝑧𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
6250, 51, 613bitri 297 . . . . . . . . . . . 12 (∀𝑞 ∈ (𝐴 × 𝐵)(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒) ↔ ∀𝑧𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
6324, 37, 623bitri 297 . . . . . . . . . . 11 (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑 ↔ ∀𝑧𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
64 frpoins3xpg.1 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → (∀𝑧𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒) → 𝜑))
6563, 64biimtrid 242 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑𝜑))
66 predeq3 6327 . . . . . . . . . . . 12 (𝑝 = ⟨𝑥, 𝑦⟩ → Pred(𝑅, (𝐴 × 𝐵), 𝑝) = Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
6766raleqdv 3324 . . . . . . . . . . 11 (𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑 ↔ ∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑))
68 sbcopeq1a 8073 . . . . . . . . . . 11 (𝑝 = ⟨𝑥, 𝑦⟩ → ([(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑𝜑))
6967, 68imbi12d 344 . . . . . . . . . 10 (𝑝 = ⟨𝑥, 𝑦⟩ → ((∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑) ↔ (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑𝜑)))
7065, 69syl5ibrcom 247 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑)))
7170ex 412 . . . . . . . 8 (𝑥𝐴 → (𝑦𝐵 → (𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑))))
727, 16, 71rexlimd 3264 . . . . . . 7 (𝑥𝐴 → (∃𝑦𝐵 𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑)))
736, 72rexlimi 3257 . . . . . 6 (∃𝑥𝐴𝑦𝐵 𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑))
741, 73sylbi 217 . . . . 5 (𝑝 ∈ (𝐴 × 𝐵) → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑))
75 fveq2 6907 . . . . . 6 (𝑝 = 𝑞 → (1st𝑝) = (1st𝑞))
76 fveq2 6907 . . . . . . 7 (𝑝 = 𝑞 → (2nd𝑝) = (2nd𝑞))
7776sbceq1d 3796 . . . . . 6 (𝑝 = 𝑞 → ([(2nd𝑝) / 𝑦]𝜑[(2nd𝑞) / 𝑦]𝜑))
7875, 77sbceqbid 3798 . . . . 5 (𝑝 = 𝑞 → ([(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑))
7974, 78frpoins2g 6368 . . . 4 ((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) → ∀𝑝 ∈ (𝐴 × 𝐵)[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑)
80 ralxpes 8160 . . . 4 (∀𝑝 ∈ (𝐴 × 𝐵)[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜑)
8179, 80sylib 218 . . 3 ((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) → ∀𝑥𝐴𝑦𝐵 𝜑)
82 frpoins3xpg.4 . . . 4 (𝑥 = 𝑋 → (𝜑𝜃))
83 frpoins3xpg.5 . . . 4 (𝑦 = 𝑌 → (𝜃𝜏))
8482, 83rspc2va 3634 . . 3 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → 𝜏)
8581, 84sylan2 593 . 2 (((𝑋𝐴𝑌𝐵) ∧ (𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵))) → 𝜏)
8685ancoms 458 1 (((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) ∧ (𝑋𝐴𝑌𝐵)) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  wral 3059  wrex 3068  [wsbc 3791  cin 3962  wss 3963  cop 4637   Po wpo 5595   Fr wfr 5638   Se wse 5639   × cxp 5687  Predcpred 6322  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-fr 5641  df-se 5642  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  xpord2indlem  8171
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