Theorem frpoins3xpg 8165

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 5709	. . . . . 6 ⊢ (𝑝 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑝 = ⟨𝑥, 𝑦⟩)
2		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑥Pred(𝑅, (𝐴 × 𝐵), 𝑝)
3		nfsbc1v 3808	. . . . . . . . 9 ⊢ Ⅎ𝑥[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑
4	2, 3	nfralw 3311	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑
5		nfsbc1v 3808	. . . . . . . 8 ⊢ Ⅎ𝑥[(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑
6	4, 5	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑥(∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → [(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑)
7		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
8		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑦Pred(𝑅, (𝐴 × 𝐵), 𝑝)
9		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(1st ‘𝑞)
10		nfsbc1v 3808	. . . . . . . . . . 11 ⊢ Ⅎ𝑦[(2nd ‘𝑞) / 𝑦]𝜑
11	9, 10	nfsbcw 3810	. . . . . . . . . 10 ⊢ Ⅎ𝑦[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑
12	8, 11	nfralw 3311	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), 𝑝)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑
13		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑦(1st ‘𝑝)
14		nfsbc1v 3808	. . . . . . . . . 10 ⊢ Ⅎ𝑦[(2nd ‘𝑝) / 𝑦]𝜑
15	13, 14	nfsbcw 3810	. . . . . . . . 9 ⊢ Ⅎ𝑦[(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑
16	12, 15	nfim 1896	. . . . . . . 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 3845	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ([(2nd ‘𝑞) / 𝑦]𝜑 ↔ [(2nd ‘𝑞) / 𝑦]𝜓))
19	18	cbvsbcvw 3822	. . . . . . . . . . . . . 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 3822	. . . . . . . . . . . . . . 15 ⊢ ([(2nd ‘𝑞) / 𝑦]𝜓 ↔ [(2nd ‘𝑞) / 𝑤]𝜒)
22	21	sbcbii 3846	. . . . . . . . . . . . . 14 ⊢ ([(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑦]𝜓 ↔ [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
23	19, 22	bitri 275	. . . . . . . . . . . . 13 ⊢ ([(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 ↔ [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
24	23	ralbii 3093	. . . . . . . . . . . 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 450	. . . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
27		predss 6329	. . . . . . . . . . . . . . . . . . 19 ⊢ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) ⊆ (𝐴 × 𝐵)
28		sseqin2 4223	. . . . . . . . . . . . . . . . . . 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 230	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) = Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
30	29	eleq2i 2833	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
31	26, 30	bitr3i 277	. . . . . . . . . . . . . . . 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 277	. . . . . . . . . . . . . 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 1819	. . . . . . . . . . . . 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 3062	. . . . . . . . . . . . 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 3062	. . . . . . . . . . . . 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 304	. . . . . . . . . . . 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 1914	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
39		nfsbc1v 3808	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧[(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒
40	38, 39	nfim 1896	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
41		nfv 1914	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤 𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)
42		nfcv 2905	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑤(1st ‘𝑞)
43		nfsbc1v 3808	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑤[(2nd ‘𝑞) / 𝑤]𝜒
44	42, 43	nfsbcw 3810	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤[(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒
45	41, 44	nfim 1896	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤(𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → [(1st ‘𝑞) / 𝑧][(2nd ‘𝑞) / 𝑤]𝜒)
46		nfv 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑞(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒)
47		eleq1 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝑞 ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)))
48		sbcopeq1a 8074	. . . . . . . . . . . . . . 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 5857	. . . . . . . . . . . . 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 3195	. . . . . . . . . . . . 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 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 5721	. . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, 𝑤⟩ ∈ ((𝐴 × 𝐵) ∩ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)) ↔ ⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
58	56, 57	bitr3i 277	. . . . . . . . . . . . . . . 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 277	. . . . . . . . . . . . . 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 1820	. . . . . . . . . . . . 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 297	. . . . . . . . . . . 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 297	. . . . . . . . . . 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 242	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∀𝑞 ∈ Pred (𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩)[(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑 → 𝜑))
66		predeq3 6325	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → Pred(𝑅, (𝐴 × 𝐵), 𝑝) = Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩))
67	66	raleqdv 3326	. . . . . . . . . . 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 8074	. . . . . . . . . . 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 247	. . . . . . . . 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 412	. . . . . . . 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 3266	. . . . . . 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 3259	. . . . . 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 217	. . . . 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 6906	. . . . . 6 ⊢ (𝑝 = 𝑞 → (1st ‘𝑝) = (1st ‘𝑞))
76		fveq2 6906	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (2nd ‘𝑝) = (2nd ‘𝑞))
77	76	sbceq1d 3793	. . . . . 6 ⊢ (𝑝 = 𝑞 → ([(2nd ‘𝑝) / 𝑦]𝜑 ↔ [(2nd ‘𝑞) / 𝑦]𝜑))
78	75, 77	sbceqbid 3795	. . . . 5 ⊢ (𝑝 = 𝑞 → ([(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑 ↔ [(1st ‘𝑞) / 𝑥][(2nd ‘𝑞) / 𝑦]𝜑))
79	74, 78	frpoins2g 6366	. . . 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 8161	. . . 4 ⊢ (∀𝑝 ∈ (𝐴 × 𝐵)[(1st ‘𝑝) / 𝑥][(2nd ‘𝑝) / 𝑦]𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
81	79, 80	sylib 218	. . 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 3634	. . 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 458	1 ⊢ (((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1538   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  [wsbc 3788   ∩ cin 3950   ⊆ wss 3951  ⟨cop 4632   Po wpo 5590   Fr wfr 5634   Se wse 5635   × cxp 5683  Predcpred 6320  ‘cfv 6561  1^st c1st 8012  2^nd c2nd 8013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-fr 5637  df-se 5638  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015

This theorem is referenced by:  xpord2indlem  8172