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Theorem txcls 21217

Description: Closure of a rectangle in the product topology. (Contributed by Mario Carneiro, 17-Sep-2015.)

**Assertion**
Ref	Expression
txcls	⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘(𝑅 ×_t 𝑆))‘(𝐴 × 𝐵)) = (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))

Proof of Theorem txcls Dummy variables 𝑠 𝑟 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 20541	. . . . . 6 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
2	1	ad2antrr 758	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝑅 ∈ Top)
3		simprl 790	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝐴 ⊆ 𝑋)
4		toponuni 20542	. . . . . . 7 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅)
5	4	ad2antrr 758	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝑋 = ∪ 𝑅)
6	3, 5	sseqtrd 3604	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝐴 ⊆ ∪ 𝑅)
7		eqid 2610	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
8	7	clscld 20661	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ⊆ ∪ 𝑅) → ((cls‘𝑅)‘𝐴) ∈ (Clsd‘𝑅))
9	2, 6, 8	syl2anc 691	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘𝑅)‘𝐴) ∈ (Clsd‘𝑅))
10		topontop 20541	. . . . . 6 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
11	10	ad2antlr 759	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝑆 ∈ Top)
12		simprr 792	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝐵 ⊆ 𝑌)
13		toponuni 20542	. . . . . . 7 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑆)
14	13	ad2antlr 759	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝑌 = ∪ 𝑆)
15	12, 14	sseqtrd 3604	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝐵 ⊆ ∪ 𝑆)
16		eqid 2610	. . . . . 6 ⊢ ∪ 𝑆 = ∪ 𝑆
17	16	clscld 20661	. . . . 5 ⊢ ((𝑆 ∈ Top ∧ 𝐵 ⊆ ∪ 𝑆) → ((cls‘𝑆)‘𝐵) ∈ (Clsd‘𝑆))
18	11, 15, 17	syl2anc 691	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘𝑆)‘𝐵) ∈ (Clsd‘𝑆))
19		txcld 21216	. . . 4 ⊢ ((((cls‘𝑅)‘𝐴) ∈ (Clsd‘𝑅) ∧ ((cls‘𝑆)‘𝐵) ∈ (Clsd‘𝑆)) → (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ∈ (Clsd‘(𝑅 ×_t 𝑆)))
20	9, 18, 19	syl2anc 691	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ∈ (Clsd‘(𝑅 ×_t 𝑆)))
21	7	sscls 20670	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ⊆ ∪ 𝑅) → 𝐴 ⊆ ((cls‘𝑅)‘𝐴))
22	2, 6, 21	syl2anc 691	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝐴 ⊆ ((cls‘𝑅)‘𝐴))
23	16	sscls 20670	. . . . 5 ⊢ ((𝑆 ∈ Top ∧ 𝐵 ⊆ ∪ 𝑆) → 𝐵 ⊆ ((cls‘𝑆)‘𝐵))
24	11, 15, 23	syl2anc 691	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝐵 ⊆ ((cls‘𝑆)‘𝐵))
25		xpss12 5148	. . . 4 ⊢ ((𝐴 ⊆ ((cls‘𝑅)‘𝐴) ∧ 𝐵 ⊆ ((cls‘𝑆)‘𝐵)) → (𝐴 × 𝐵) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
26	22, 24, 25	syl2anc 691	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → (𝐴 × 𝐵) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
27		eqid 2610	. . . 4 ⊢ ∪ (𝑅 ×_t 𝑆) = ∪ (𝑅 ×_t 𝑆)
28	27	clsss2 20686	. . 3 ⊢ (((((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ∈ (Clsd‘(𝑅 ×_t 𝑆)) ∧ (𝐴 × 𝐵) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵))) → ((cls‘(𝑅 ×_t 𝑆))‘(𝐴 × 𝐵)) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
29	20, 26, 28	syl2anc 691	. 2 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘(𝑅 ×_t 𝑆))‘(𝐴 × 𝐵)) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
30		relxp 5150	. . . 4 ⊢ Rel (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵))
31	30	a1i 11	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → Rel (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
32		opelxp 5070	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ↔ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵)))
33		eltx 21181	. . . . . . . . 9 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑢 ∈ (𝑅 ×_t 𝑆) ↔ ∀𝑧 ∈ 𝑢 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢)))
34	33	ad2antrr 758	. . . . . . . 8 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝑢 ∈ (𝑅 ×_t 𝑆) ↔ ∀𝑧 ∈ 𝑢 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢)))
35		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝑟 × 𝑠) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠)))
36	35	anbi1d 737	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢)))
37	36	2rexbidv 3039	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢)))
38	37	rspccva 3281	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝑢 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑢) → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))
39	2	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑅 ∈ Top)
40	6	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝐴 ⊆ ∪ 𝑅)
41		simplrl 796	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑥 ∈ ((cls‘𝑅)‘𝐴))
42		simprll 798	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑟 ∈ 𝑅)
43		simprrl 800	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → ⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠))
44		opelxp 5070	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ↔ (𝑥 ∈ 𝑟 ∧ 𝑦 ∈ 𝑠))
45	43, 44	sylib 207	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → (𝑥 ∈ 𝑟 ∧ 𝑦 ∈ 𝑠))
46	45	simpld 474	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑥 ∈ 𝑟)
47	7	clsndisj 20689	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ⊆ ∪ 𝑅 ∧ 𝑥 ∈ ((cls‘𝑅)‘𝐴)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑥 ∈ 𝑟)) → (𝑟 ∩ 𝐴) ≠ ∅)
48	39, 40, 41, 42, 46, 47	syl32anc 1326	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → (𝑟 ∩ 𝐴) ≠ ∅)
49		n0 3890	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∩ 𝐴) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑟 ∩ 𝐴))
50	48, 49	sylib 207	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → ∃𝑧 𝑧 ∈ (𝑟 ∩ 𝐴))
51	11	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑆 ∈ Top)
52	15	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝐵 ⊆ ∪ 𝑆)
53		simplrr 797	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑦 ∈ ((cls‘𝑆)‘𝐵))
54		simprlr 799	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑠 ∈ 𝑆)
55	45	simprd 478	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑦 ∈ 𝑠)
56	16	clsndisj 20689	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ Top ∧ 𝐵 ⊆ ∪ 𝑆 ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ 𝑠)) → (𝑠 ∩ 𝐵) ≠ ∅)
57	51, 52, 53, 54, 55, 56	syl32anc 1326	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → (𝑠 ∩ 𝐵) ≠ ∅)
58		n0 3890	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∩ 𝐵) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑠 ∩ 𝐵))
59	57, 58	sylib 207	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → ∃𝑤 𝑤 ∈ (𝑠 ∩ 𝐵))
60		eeanv 2170	. . . . . . . . . . . . . 14 ⊢ (∃𝑧∃𝑤(𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵)) ↔ (∃𝑧 𝑧 ∈ (𝑟 ∩ 𝐴) ∧ ∃𝑤 𝑤 ∈ (𝑠 ∩ 𝐵)))
61		inss1 3795	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) ⊆ (𝑟 × 𝑠)
62		opelxpi 5072	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵)) → ⟨𝑧, 𝑤⟩ ∈ ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))
63		inxp 5176	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))
64	62, 63	syl6eleqr 2699	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵)) → ⟨𝑧, 𝑤⟩ ∈ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
65	61, 64	sseldi 3566	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵)) → ⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠))
66		simprrr 801	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → (𝑟 × 𝑠) ⊆ 𝑢)
67	66	sselda 3568	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠)) → ⟨𝑧, 𝑤⟩ ∈ 𝑢)
68	65, 67	sylan2 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) ∧ (𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵))) → ⟨𝑧, 𝑤⟩ ∈ 𝑢)
69		inss2 3796	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
70	69, 64	sseldi 3566	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵)) → ⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵))
71	70	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) ∧ (𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵))) → ⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵))
72		inelcm 3984	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑢 ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵)) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)
73	68, 71, 72	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) ∧ (𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵))) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)
74	73	ex 449	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → ((𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵)) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
75	74	exlimdvv 1849	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → (∃𝑧∃𝑤(𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵)) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
76	60, 75	syl5bir 232	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → ((∃𝑧 𝑧 ∈ (𝑟 ∩ 𝐴) ∧ ∃𝑤 𝑤 ∈ (𝑠 ∩ 𝐵)) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
77	50, 59, 76	mp2and 711	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)
78	77	expr 641	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
79	78	rexlimdvva 3020	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
80	38, 79	syl5 33	. . . . . . . . 9 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ((∀𝑧 ∈ 𝑢 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑢) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
81	80	expd 451	. . . . . . . 8 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (∀𝑧 ∈ 𝑢 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) → (⟨𝑥, 𝑦⟩ ∈ 𝑢 → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)))
82	34, 81	sylbid 229	. . . . . . 7 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝑢 ∈ (𝑅 ×_t 𝑆) → (⟨𝑥, 𝑦⟩ ∈ 𝑢 → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)))
83	82	ralrimiv 2948	. . . . . 6 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ∀𝑢 ∈ (𝑅 ×_t 𝑆)(⟨𝑥, 𝑦⟩ ∈ 𝑢 → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
84		txtopon 21204	. . . . . . . . 9 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
85	84	ad2antrr 758	. . . . . . . 8 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
86		topontop 20541	. . . . . . . 8 ⊢ ((𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑅 ×_t 𝑆) ∈ Top)
87	85, 86	syl 17	. . . . . . 7 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝑅 ×_t 𝑆) ∈ Top)
88		xpss12 5148	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
89	88	ad2antlr 759	. . . . . . . 8 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
90		toponuni 20542	. . . . . . . . 9 ⊢ ((𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = ∪ (𝑅 ×_t 𝑆))
91	85, 90	syl 17	. . . . . . . 8 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝑋 × 𝑌) = ∪ (𝑅 ×_t 𝑆))
92	89, 91	sseqtrd 3604	. . . . . . 7 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝐴 × 𝐵) ⊆ ∪ (𝑅 ×_t 𝑆))
93	7	clsss3 20673	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ⊆ ∪ 𝑅) → ((cls‘𝑅)‘𝐴) ⊆ ∪ 𝑅)
94	2, 6, 93	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘𝑅)‘𝐴) ⊆ ∪ 𝑅)
95	94, 5	sseqtr4d 3605	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘𝑅)‘𝐴) ⊆ 𝑋)
96	95	sselda 3568	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑥 ∈ ((cls‘𝑅)‘𝐴)) → 𝑥 ∈ 𝑋)
97	96	adantrr 749	. . . . . . . . 9 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → 𝑥 ∈ 𝑋)
98	16	clsss3 20673	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Top ∧ 𝐵 ⊆ ∪ 𝑆) → ((cls‘𝑆)‘𝐵) ⊆ ∪ 𝑆)
99	11, 15, 98	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘𝑆)‘𝐵) ⊆ ∪ 𝑆)
100	99, 14	sseqtr4d 3605	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘𝑆)‘𝐵) ⊆ 𝑌)
101	100	sselda 3568	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵)) → 𝑦 ∈ 𝑌)
102	101	adantrl 748	. . . . . . . . 9 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → 𝑦 ∈ 𝑌)
103		opelxpi 5072	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
104	97, 102, 103	syl2anc 691	. . . . . . . 8 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
105	104, 91	eleqtrd 2690	. . . . . . 7 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ⟨𝑥, 𝑦⟩ ∈ ∪ (𝑅 ×_t 𝑆))
106	27	elcls 20687	. . . . . . 7 ⊢ (((𝑅 ×_t 𝑆) ∈ Top ∧ (𝐴 × 𝐵) ⊆ ∪ (𝑅 ×_t 𝑆) ∧ ⟨𝑥, 𝑦⟩ ∈ ∪ (𝑅 ×_t 𝑆)) → (⟨𝑥, 𝑦⟩ ∈ ((cls‘(𝑅 ×_t 𝑆))‘(𝐴 × 𝐵)) ↔ ∀𝑢 ∈ (𝑅 ×_t 𝑆)(⟨𝑥, 𝑦⟩ ∈ 𝑢 → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)))
107	87, 92, 105, 106	syl3anc 1318	. . . . . 6 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (⟨𝑥, 𝑦⟩ ∈ ((cls‘(𝑅 ×_t 𝑆))‘(𝐴 × 𝐵)) ↔ ∀𝑢 ∈ (𝑅 ×_t 𝑆)(⟨𝑥, 𝑦⟩ ∈ 𝑢 → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)))
108	83, 107	mpbird 246	. . . . 5 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ⟨𝑥, 𝑦⟩ ∈ ((cls‘(𝑅 ×_t 𝑆))‘(𝐴 × 𝐵)))
109	108	ex 449	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵)) → ⟨𝑥, 𝑦⟩ ∈ ((cls‘(𝑅 ×_t 𝑆))‘(𝐴 × 𝐵))))
110	32, 109	syl5bi 231	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → (⟨𝑥, 𝑦⟩ ∈ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) → ⟨𝑥, 𝑦⟩ ∈ ((cls‘(𝑅 ×_t 𝑆))‘(𝐴 × 𝐵))))
111	31, 110	relssdv 5135	. 2 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ⊆ ((cls‘(𝑅 ×_t 𝑆))‘(𝐴 × 𝐵)))
112	29, 111	eqssd 3585	1 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘(𝑅 ×_t 𝑆))‘(𝐴 × 𝐵)) = (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ⟨cop 4131 ∪ cuni 4372 × cxp 5036 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 Topctop 20517 TopOnctopon 20518 Clsdccld 20630 clsccl 20632 ×_t ctx 21173

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635 df-tx 21175

This theorem is referenced by: clssubg 21722

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