Theorem frpoins3xpg 8135

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 5671	. . . . . 6 ⊢ (𝑝 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑝 = ⟨𝑥, 𝑦⟩)
2		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑥Pred(𝑅, (𝐴 × 𝐵), 𝑝)
3		nfsbc1v 3758	. . . . . . . . 9 ⊢ Ⅎ𝑥[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑
4	2, 3	nfralw 3309	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑
5		nfsbc1v 3758	. . . . . . . 8 ⊢ Ⅎ𝑥[(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑
6	4, 5	nfim 1929	. . . . . . 7 ⊢ Ⅎ𝑥(∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → [(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑)
7		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
8		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑦Pred(𝑅, (𝐴 × 𝐵), 𝑝)
9		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(1st ‘𝑞)
10		nfsbc1v 3758	. . . . . . . . . . 11 ⊢ Ⅎ𝑦[(2nd ‘𝑞) / 𝑦]𝜑
11	9, 10	nfsbcw 3760	. . . . . . . . . 10 ⊢ Ⅎ𝑦[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑
12	8, 11	nfralw 3309	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑
13		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑦(1st ‘𝑝)
14		nfsbc1v 3758	. . . . . . . . . 10 ⊢ Ⅎ𝑦[(2nd ‘𝑝) / 𝑦]𝜑
15	13, 14	nfsbcw 3760	. . . . . . . . 9 ⊢ Ⅎ𝑦[(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑
16	12, 15	nfim 1929	. . . . . . . 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 3793	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ([(2nd ‘𝑞) / 𝑦]𝜑 ↔ [(2nd ‘𝑞) / 𝑦]𝜓))
19	18	cbvsbcvw 3772	. . . . . . . . . . . . . 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 3772	. . . . . . . . . . . . . . 15 ⊢ ([(2nd ‘𝑞) / 𝑦]𝜓 ↔ [(2nd ‘𝑞) / 𝑤]𝜒)
22	21	sbcbii 3794	. . . . . . . . . . . . . 14 ⊢ ([(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑦]𝜓 ↔ [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
23	19, 22	bitri 278	. . . . . . . . . . . . 13 ⊢ ([(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 ↔ [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
24	23	ralbii 3108	. . . . . . . . . . . 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 456	. . . . . . . . . . . . . . 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 3914	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
27		predss 6301	. . . . . . . . . . . . . . . . . . 19 ⊢ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) ⊆ (𝐴 × 𝐵)
28		sseqin2 4168	. . . . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) = Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
30	29	eleq2i 2852	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
31	26, 30	bitr3i 280	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
32	31	imbi1i 352	. . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . 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 1852	. . . . . . . . . . . . 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 3077	. . . . . . . . . . . . 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 3077	. . . . . . . . . . . . 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 307	. . . . . . . . . . . 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 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
39		nfsbc1v 3758	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧[(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒
40	38, 39	nfim 1929	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
41		nfv 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
42		nfcv 2922	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑤(1st ‘𝑞)
43		nfsbc1v 3758	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑤[(2nd ‘𝑞) / 𝑤]𝜒
44	42, 43	nfsbcw 3760	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤[(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒
45	41, 44	nfim 1929	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
46		nfv 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑞(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)
47		eleq1 2848	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
48		sbcopeq1a 8043	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → ([(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒 ↔ 𝜒))
49	47, 48	imbi12d 347	. . . . . . . . . . . . . 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 5820	. . . . . . . . . . . . 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 3198	. . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . . 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 5683	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))
54	53	imbi1i 352	. . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . 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 3914	. . . . . . . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, 𝑤⟩ ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
58	56, 57	bitr3i 280	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
59	58	imbi1i 352	. . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . 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 1853	. . . . . . . . . . . . 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 300	. . . . . . . . . . . 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 300	. . . . . . . . . . 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 245	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → 𝜑))
66		predeq3 6297	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → Pred(𝑅, (𝐴 × 𝐵), 𝑝) = Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
67	66	raleqdv 3319	. . . . . . . . . . 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 8043	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑 ↔ 𝜑))
69	67, 68	imbi12d 347	. . . . . . . . . 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 250	. . . . . . . . 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 418	. . . . . . . 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 3269	. . . . . . 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 3262	. . . . . 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 220	. . . . 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 6873	. . . . . 6 ⊢ (𝑝 = 𝑞 → (1st ‘𝑝) = (1st ‘𝑞))
76		fveq2 6873	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (2nd ‘𝑝) = (2nd ‘𝑞))
77	76	sbceq1d 3743	. . . . . 6 ⊢ (𝑝 = 𝑞 → ([(2nd ‘𝑝) / 𝑦]𝜑 ↔ [(2nd ‘𝑞) / 𝑦]𝜑))
78	75, 77	sbceqbid 3745	. . . . 5 ⊢ (𝑝 = 𝑞 → ([(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑 ↔ [(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑))
79	74, 78	frpoins2g 6337	. . . 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 8131	. . . 4 ⊢ (∀𝑝 ∈ (𝐴 × 𝐵)[(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
81	79, 80	sylib 221	. . 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 3587	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑) → 𝜏)
85	81, 84	sylan2 605	. 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵))) → 𝜏)
86	85	ancoms 464	1 ⊢ (((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  [wsbc 3738   ∩ cin 3897   ⊆ wss 3898  ⟨cop 4589   Po wpo 5553   Fr wfr 5597   Se wse 5598   × cxp 5645  Predcpred 6292  ‘cfv 6527  1^st c1st 7982  2^nd c2nd 7983

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-po 5555  df-fr 5600  df-se 5601  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-iota 6483  df-fun 6529  df-fv 6535  df-1st 7984  df-2nd 7985

This theorem is used by:  xpord2indlem  8142