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Theorem frpoins3xpg 33390
Description: Special case of founded partial recursion over a cross 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:   𝑤,𝐴,𝑥,𝑦,𝑧   𝑤,𝐵,𝑥,𝑦,𝑧   𝜒,𝑦   𝜑,𝑧   𝜓,𝑤,𝑥   𝑤,𝑅,𝑥,𝑦,𝑧   𝜏,𝑦   𝜃,𝑥   𝑥,𝑤,𝑦,𝑧   𝑥,𝑋,𝑦   𝑥,𝑧   𝑦,𝑌   𝑦,𝑧
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 5549 . . . . . 6 (𝑝 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑝 = ⟨𝑥, 𝑦⟩)
2 nfcv 2899 . . . . . . . . 9 𝑥Pred(𝑅, (𝐴 × 𝐵), 𝑝)
3 nfsbc1v 3700 . . . . . . . . 9 𝑥[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑
42, 3nfralw 3138 . . . . . . . 8 𝑥𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑
5 nfsbc1v 3700 . . . . . . . 8 𝑥[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑
64, 5nfim 1903 . . . . . . 7 𝑥(∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑)
7 nfv 1921 . . . . . . . 8 𝑦 𝑥𝐴
8 nfcv 2899 . . . . . . . . . 10 𝑦Pred(𝑅, (𝐴 × 𝐵), 𝑝)
9 nfcv 2899 . . . . . . . . . . 11 𝑦(1st𝑞)
10 nfsbc1v 3700 . . . . . . . . . . 11 𝑦[(2nd𝑞) / 𝑦]𝜑
119, 10nfsbcw 3702 . . . . . . . . . 10 𝑦[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑
128, 11nfralw 3138 . . . . . . . . 9 𝑦𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑
13 nfcv 2899 . . . . . . . . . 10 𝑦(1st𝑝)
14 nfsbc1v 3700 . . . . . . . . . 10 𝑦[(2nd𝑝) / 𝑦]𝜑
1513, 14nfsbcw 3702 . . . . . . . . 9 𝑦[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑
1612, 15nfim 1903 . . . . . . . 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 3736 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ([(2nd𝑞) / 𝑦]𝜑[(2nd𝑞) / 𝑦]𝜓))
1918cbvsbcvw 3715 . . . . . . . . . . . . . 14 ([(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑦]𝜓)
20 frpoins3xpg.3 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑤 → (𝜓𝜒))
2120cbvsbcvw 3715 . . . . . . . . . . . . . . 15 ([(2nd𝑞) / 𝑦]𝜓[(2nd𝑞) / 𝑤]𝜒)
2221sbcbii 3738 . . . . . . . . . . . . . 14 ([(1st𝑞) / 𝑧][(2nd𝑞) / 𝑦]𝜓[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)
2319, 22bitri 278 . . . . . . . . . . . . 13 ([(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)
2423ralbii 3080 . . . . . . . . . . . 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 454 . . . . . . . . . . . . . . 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 3859 . . . . . . . . . . . . . . . . 17 (𝑞 ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
27 predss 6136 . . . . . . . . . . . . . . . . . . 19 Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) ⊆ (𝐴 × 𝐵)
28 sseqin2 4106 . . . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . . . . 18 ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) = Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
3029eleq2i 2824 . . . . . . . . . . . . . . . . 17 (𝑞 ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
3126, 30bitr3i 280 . . . . . . . . . . . . . . . 16 ((𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
3231imbi1i 353 . . . . . . . . . . . . . . 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 280 . . . . . . . . . . . . . 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 1826 . . . . . . . . . . . . 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 3058 . . . . . . . . . . . . 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 3058 . . . . . . . . . . . . 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 307 . . . . . . . . . . . 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 1921 . . . . . . . . . . . . . . 15 𝑧 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
39 nfsbc1v 3700 . . . . . . . . . . . . . . 15 𝑧[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒
4038, 39nfim 1903 . . . . . . . . . . . . . 14 𝑧(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)
41 nfv 1921 . . . . . . . . . . . . . . 15 𝑤 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
42 nfcv 2899 . . . . . . . . . . . . . . . 16 𝑤(1st𝑞)
43 nfsbc1v 3700 . . . . . . . . . . . . . . . 16 𝑤[(2nd𝑞) / 𝑤]𝜒
4442, 43nfsbcw 3702 . . . . . . . . . . . . . . 15 𝑤[(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒
4541, 44nfim 1903 . . . . . . . . . . . . . 14 𝑤(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒)
46 nfv 1921 . . . . . . . . . . . . . 14 𝑞(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)
47 eleq1 2820 . . . . . . . . . . . . . . 15 (𝑞 = ⟨𝑧, 𝑤⟩ → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
48 sbcopeq1a 7773 . . . . . . . . . . . . . . 15 (𝑞 = ⟨𝑧, 𝑤⟩ → ([(1st𝑞) / 𝑧][(2nd𝑞) / 𝑤]𝜒𝜒))
4947, 48imbi12d 348 . . . . . . . . . . . . . 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 5689 . . . . . . . . . . . . 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 3113 . . . . . . . . . . . . 13 (∀𝑧𝐴𝑤𝐵 (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒) ↔ ∀𝑧𝑤((𝑧𝐴𝑤𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)))
52 impexp 454 . . . . . . . . . . . . . . . 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 5561 . . . . . . . . . . . . . . . . 17 (⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ↔ (𝑧𝐴𝑤𝐵))
5453imbi1i 353 . . . . . . . . . . . . . . . 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 278 . . . . . . . . . . . . . . 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 3859 . . . . . . . . . . . . . . . . 17 (⟨𝑧, 𝑤⟩ ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ (⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
5729eleq2i 2824 . . . . . . . . . . . . . . . . 17 (⟨𝑧, 𝑤⟩ ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
5856, 57bitr3i 280 . . . . . . . . . . . . . . . 16 ((⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
5958imbi1i 353 . . . . . . . . . . . . . . 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 280 . . . . . . . . . . . . . 14 (((𝑧𝐴𝑤𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)) ↔ (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
61602albii 1827 . . . . . . . . . . . . 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 300 . . . . . . . . . . . 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 300 . . . . . . . . . . 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, 64syl5bi 245 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑𝜑))
66 predeq3 6133 . . . . . . . . . . . 12 (𝑝 = ⟨𝑥, 𝑦⟩ → Pred(𝑅, (𝐴 × 𝐵), 𝑝) = Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
6766raleqdv 3316 . . . . . . . . . . 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 7773 . . . . . . . . . . 11 (𝑝 = ⟨𝑥, 𝑦⟩ → ([(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑𝜑))
6967, 68imbi12d 348 . . . . . . . . . 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 250 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑)))
7170ex 416 . . . . . . . 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 3227 . . . . . . 7 (𝑥𝐴 → (∃𝑦𝐵 𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑)))
736, 72rexlimi 3225 . . . . . 6 (∃𝑥𝐴𝑦𝐵 𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑))
741, 73sylbi 220 . . . . 5 (𝑝 ∈ (𝐴 × 𝐵) → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑))
75 fveq2 6674 . . . . . 6 (𝑝 = 𝑞 → (1st𝑝) = (1st𝑞))
76 fveq2 6674 . . . . . . 7 (𝑝 = 𝑞 → (2nd𝑝) = (2nd𝑞))
7776sbceq1d 3685 . . . . . 6 (𝑝 = 𝑞 → ([(2nd𝑝) / 𝑦]𝜑[(2nd𝑞) / 𝑦]𝜑))
7875, 77sbceqbid 3687 . . . . 5 (𝑝 = 𝑞 → ([(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑[(1st𝑞) / 𝑥][(2nd𝑞) / 𝑦]𝜑))
7974, 78frpoins2g 33388 . . . 4 ((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) → ∀𝑝 ∈ (𝐴 × 𝐵)[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑)
80 ralxpes 33252 . . . 4 (∀𝑝 ∈ (𝐴 × 𝐵)[(1st𝑝) / 𝑥][(2nd𝑝) / 𝑦]𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜑)
8179, 80sylib 221 . . 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 3537 . . 3 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → 𝜏)
8581, 84sylan2 596 . 2 (((𝑋𝐴𝑌𝐵) ∧ (𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵))) → 𝜏)
8685ancoms 462 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 209  wa 399  w3a 1088  wal 1540   = wceq 1542  wcel 2114  wral 3053  wrex 3054  [wsbc 3680  cin 3842  wss 3843  cop 4522   Po wpo 5440   Fr wfr 5480   Se wse 5481   × cxp 5523  Predcpred 6128  cfv 6339  1st c1st 7712  2nd c2nd 7713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-po 5442  df-fr 5483  df-se 5484  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-iota 6297  df-fun 6341  df-fv 6347  df-1st 7714  df-2nd 7715
This theorem is referenced by:  xpord2ind  33407
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