Theorem frpoins3xpg 8072

Description: Special case of founded partial induction over a Cartesian product. (Contributed by Scott Fenton, 22-Aug-2024.)

Hypotheses

Ref	Expression
frpoins3xpg.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒) → 𝜑))
frpoins3xpg.2	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
frpoins3xpg.3	⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))
frpoins3xpg.4	⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜃))
frpoins3xpg.5	⊢ (𝑦 = 𝑌 → (𝜃 ↔ 𝜏))

Assertion

Ref	Expression
frpoins3xpg	⊢ (((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 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.

Step	Hyp	Ref	Expression
1		elxp2 5657	. . . . . 6 ⊢ (𝑝 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑝 = ⟨𝑥, 𝑦⟩)
2		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑥Pred(𝑅, (𝐴 × 𝐵), 𝑝)
3		nfsbc1v 3759	. . . . . . . . 9 ⊢ Ⅎ𝑥[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑
4	2, 3	nfralw 3294	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑
5		nfsbc1v 3759	. . . . . . . 8 ⊢ Ⅎ𝑥[(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑
6	4, 5	nfim 1899	. . . . . . 7 ⊢ Ⅎ𝑥(∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → [(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑)
7		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
8		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑦Pred(𝑅, (𝐴 × 𝐵), 𝑝)
9		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(1st ‘𝑞)
10		nfsbc1v 3759	. . . . . . . . . . 11 ⊢ Ⅎ𝑦[(2nd ‘𝑞) / 𝑦]𝜑
11	9, 10	nfsbcw 3761	. . . . . . . . . 10 ⊢ Ⅎ𝑦[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑
12	8, 11	nfralw 3294	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑
13		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑦(1st ‘𝑝)
14		nfsbc1v 3759	. . . . . . . . . 10 ⊢ Ⅎ𝑦[(2nd ‘𝑝) / 𝑦]𝜑
15	13, 14	nfsbcw 3761	. . . . . . . . 9 ⊢ Ⅎ𝑦[(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑
16	12, 15	nfim 1899	. . . . . . . 8 ⊢ Ⅎ𝑦(∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → [(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑)
17		frpoins3xpg.2	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
18	17	sbcbidv 3798	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ([(2nd ‘𝑞) / 𝑦]𝜑 ↔ [(2nd ‘𝑞) / 𝑦]𝜓))
19	18	cbvsbcvw 3774	. . . . . . . . . . . . . 14 ⊢ ([(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 ↔ [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑦]𝜓)
20		frpoins3xpg.3	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))
21	20	cbvsbcvw 3774	. . . . . . . . . . . . . . 15 ⊢ ([(2nd ‘𝑞) / 𝑦]𝜓 ↔ [(2nd ‘𝑞) / 𝑤]𝜒)
22	21	sbcbii 3799	. . . . . . . . . . . . . 14 ⊢ ([(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑦]𝜓 ↔ [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
23	19, 22	bitri 274	. . . . . . . . . . . . 13 ⊢ ([(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 ↔ [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
24	23	ralbii 3096	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 ↔ ∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
25		impexp 451	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒) ↔ (𝑞 ∈ (𝐴 × 𝐵) → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)))
26		elin 3926	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
27		predss 6261	. . . . . . . . . . . . . . . . . . 19 ⊢ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) ⊆ (𝐴 × 𝐵)
28		sseqin2 4175	. . . . . . . . . . . . . . . . . . 19 ⊢ (Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) ⊆ (𝐴 × 𝐵) ↔ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) = Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
29	27, 28	mpbi 229	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) = Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
30	29	eleq2i 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
31	26, 30	bitr3i 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
32	31	imbi1i 349	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒) ↔ (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒))
33	25, 32	bitr3i 276	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ (𝐴 × 𝐵) → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)) ↔ (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒))
34	33	albii 1821	. . . . . . . . . . . . 13 ⊢ (∀𝑞(𝑞 ∈ (𝐴 × 𝐵) → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)) ↔ ∀𝑞(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒))
35		df-ral 3065	. . . . . . . . . . . . 13 ⊢ (∀𝑞 ∈ (𝐴 × 𝐵)(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒) ↔ ∀𝑞(𝑞 ∈ (𝐴 × 𝐵) → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)))
36		df-ral 3065	. . . . . . . . . . . . 13 ⊢ (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒 ↔ ∀𝑞(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒))
37	34, 35, 36	3bitr4ri 303	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒 ↔ ∀𝑞 ∈ (𝐴 × 𝐵)(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒))
38		nfv 1917	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
39		nfsbc1v 3759	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧[(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒
40	38, 39	nfim 1899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
41		nfv 1917	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
42		nfcv 2907	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑤(1st ‘𝑞)
43		nfsbc1v 3759	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑤[(2nd ‘𝑞) / 𝑤]𝜒
44	42, 43	nfsbcw 3761	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤[(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒
45	41, 44	nfim 1899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
46		nfv 1917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑞(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)
47		eleq1 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
48		sbcopeq1a 7981	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → ([(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒 ↔ 𝜒))
49	47, 48	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → ((𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒) ↔ (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)))
50	40, 45, 46, 49	ralxpf 5802	. . . . . . . . . . . . 13 ⊢ (∀𝑞 ∈ (𝐴 × 𝐵)(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
51		r2al 3191	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒) ↔ ∀𝑧∀𝑤((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)))
52		impexp 451	. . . . . . . . . . . . . . . 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 5669	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))
54	53	imbi1i 349	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)))
55	52, 54	bitri 274	. . . . . . . . . . . . . . 15 ⊢ (((⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) → 𝜒) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)))
56		elin 3926	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, 𝑤⟩ ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ (⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
57	29	eleq2i 2829	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, 𝑤⟩ ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
58	56, 57	bitr3i 276	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
59	58	imbi1i 349	. . . . . . . . . . . . . . 15 ⊢ (((⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) → 𝜒) ↔ (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
60	55, 59	bitr3i 276	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)) ↔ (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
61	60	2albii 1822	. . . . . . . . . . . . 13 ⊢ (∀𝑧∀𝑤((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → (⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)) ↔ ∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
62	50, 51, 61	3bitri 296	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ (𝐴 × 𝐵)(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒) ↔ ∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
63	24, 37, 62	3bitri 296	. . . . . . . . . . 11 ⊢ (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 ↔ ∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒))
64		frpoins3xpg.1	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒) → 𝜑))
65	63, 64	biimtrid 241	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → 𝜑))
66		predeq3 6257	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → Pred(𝑅, (𝐴 × 𝐵), 𝑝) = Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
67	66	raleqdv 3313	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 ↔ ∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑))
68		sbcopeq1a 7981	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑 ↔ 𝜑))
69	67, 68	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → [(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑) ↔ (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → 𝜑)))
70	65, 69	syl5ibrcom 246	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → [(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑)))
71	70	ex 413	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → [(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑))))
72	7, 16, 71	rexlimd 3249	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → [(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑)))
73	6, 72	rexlimi 3242	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑝 = ⟨𝑥, 𝑦⟩ → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → [(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑))
74	1, 73	sylbi 216	. . . . 5 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → [(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑))
75		fveq2 6842	. . . . . 6 ⊢ (𝑝 = 𝑞 → (1st ‘𝑝) = (1st ‘𝑞))
76		fveq2 6842	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (2nd ‘𝑝) = (2nd ‘𝑞))
77	76	sbceq1d 3744	. . . . . 6 ⊢ (𝑝 = 𝑞 → ([(2nd ‘𝑝) / 𝑦]𝜑 ↔ [(2nd ‘𝑞) / 𝑦]𝜑))
78	75, 77	sbceqbid 3746	. . . . 5 ⊢ (𝑝 = 𝑞 → ([(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑 ↔ [(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑))
79	74, 78	frpoins2g 6299	. . . 4 ⊢ ((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) → ∀𝑝 ∈ (𝐴 × 𝐵)[(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑)
80		ralxpes 8068	. . . 4 ⊢ (∀𝑝 ∈ (𝐴 × 𝐵)[(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
81	79, 80	sylib 217	. . 3 ⊢ ((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
82		frpoins3xpg.4	. . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜃))
83		frpoins3xpg.5	. . . 4 ⊢ (𝑦 = 𝑌 → (𝜃 ↔ 𝜏))
84	82, 83	rspc2va 3591	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑) → 𝜏)
85	81, 84	sylan2 593	. 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵))) → 𝜏)
86	85	ancoms 459	1 ⊢ (((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  [wsbc 3739   ∩ cin 3909   ⊆ wss 3910  ⟨cop 4592   Po wpo 5543   Fr wfr 5585   Se wse 5586   × cxp 5631  Predcpred 6252  ‘cfv 6496  1^st c1st 7919  2^nd c2nd 7920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-fr 5588  df-se 5589  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-iota 6448  df-fun 6498  df-fv 6504  df-1st 7921  df-2nd 7922

This theorem is referenced by:  xpord2indlem  8079