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Theorem txcls 23565

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 22874	. . . . . 6 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
2	1	ad2antrr 727	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝑅 ∈ Top)
3		simprl 771	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝐴 ⊆ 𝑋)
4		toponuni 22875	. . . . . . 7 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅)
5	4	ad2antrr 727	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝑋 = ∪ 𝑅)
6	3, 5	sseqtrd 3972	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝐴 ⊆ ∪ 𝑅)
7		eqid 2737	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
8	7	clscld 23008	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ⊆ ∪ 𝑅) → ((cls‘𝑅)‘𝐴) ∈ (Clsd‘𝑅))
9	2, 6, 8	syl2anc 585	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘𝑅)‘𝐴) ∈ (Clsd‘𝑅))
10		topontop 22874	. . . . . 6 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
11	10	ad2antlr 728	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝑆 ∈ Top)
12		simprr 773	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝐵 ⊆ 𝑌)
13		toponuni 22875	. . . . . . 7 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑆)
14	13	ad2antlr 728	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝑌 = ∪ 𝑆)
15	12, 14	sseqtrd 3972	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝐵 ⊆ ∪ 𝑆)
16		eqid 2737	. . . . . 6 ⊢ ∪ 𝑆 = ∪ 𝑆
17	16	clscld 23008	. . . . 5 ⊢ ((𝑆 ∈ Top ∧ 𝐵 ⊆ ∪ 𝑆) → ((cls‘𝑆)‘𝐵) ∈ (Clsd‘𝑆))
18	11, 15, 17	syl2anc 585	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘𝑆)‘𝐵) ∈ (Clsd‘𝑆))
19		txcld 23564	. . . 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 585	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ∈ (Clsd‘(𝑅 ×_t 𝑆)))
21	7	sscls 23017	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ⊆ ∪ 𝑅) → 𝐴 ⊆ ((cls‘𝑅)‘𝐴))
22	2, 6, 21	syl2anc 585	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝐴 ⊆ ((cls‘𝑅)‘𝐴))
23	16	sscls 23017	. . . . 5 ⊢ ((𝑆 ∈ Top ∧ 𝐵 ⊆ ∪ 𝑆) → 𝐵 ⊆ ((cls‘𝑆)‘𝐵))
24	11, 15, 23	syl2anc 585	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → 𝐵 ⊆ ((cls‘𝑆)‘𝐵))
25		xpss12 5649	. . . 4 ⊢ ((𝐴 ⊆ ((cls‘𝑅)‘𝐴) ∧ 𝐵 ⊆ ((cls‘𝑆)‘𝐵)) → (𝐴 × 𝐵) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
26	22, 24, 25	syl2anc 585	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → (𝐴 × 𝐵) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
27		eqid 2737	. . . 4 ⊢ ∪ (𝑅 ×_t 𝑆) = ∪ (𝑅 ×_t 𝑆)
28	27	clsss2 23033	. . 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 585	. 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 5652	. . . 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 5670	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ↔ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵)))
33		eltx 23529	. . . . . . . . 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 727	. . . . . . . 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 2825	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝑟 × 𝑠) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠)))
36	35	anbi1d 632	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢)))
37	36	2rexbidv 3203	. . . . . . . . . . 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 3577	. . . . . . . . . 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 727	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . 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 5670	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ↔ (𝑥 ∈ 𝑟 ∧ 𝑦 ∈ 𝑠))
45	43, 44	sylib 218	. . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . 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 23036	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ⊆ ∪ 𝑅 ∧ 𝑥 ∈ ((cls‘𝑅)‘𝐴)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑥 ∈ 𝑟)) → (𝑟 ∩ 𝐴) ≠ ∅)
48	39, 40, 41, 42, 46, 47	syl32anc 1381	. . . . . . . . . . . . . 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 4307	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∩ 𝐴) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑟 ∩ 𝐴))
50	48, 49	sylib 218	. . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . 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 23036	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ Top ∧ 𝐵 ⊆ ∪ 𝑆 ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ 𝑠)) → (𝑠 ∩ 𝐵) ≠ ∅)
57	51, 52, 53, 54, 55, 56	syl32anc 1381	. . . . . . . . . . . . . 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 4307	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∩ 𝐵) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑠 ∩ 𝐵))
59	57, 58	sylib 218	. . . . . . . . . . . . 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		exdistrv 1957	. . . . . . . . . . . . . 14 ⊢ (∃𝑧∃𝑤(𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵)) ↔ (∃𝑧 𝑧 ∈ (𝑟 ∩ 𝐴) ∧ ∃𝑤 𝑤 ∈ (𝑠 ∩ 𝐵)))
61		opelxpi 5671	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵)) → ⟨𝑧, 𝑤⟩ ∈ ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))
62		inxp 5790	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))
63	61, 62	eleqtrrdi 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵)) → ⟨𝑧, 𝑤⟩ ∈ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
64	63	elin1d 4158	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵)) → ⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠))
65		simprrr 782	. . . . . . . . . . . . . . . . . . 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)
66	65	sselda 3935	. . . . . . . . . . . . . . . . . 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)
67	64, 66	sylan2 594	. . . . . . . . . . . . . . . . 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)
68	63	elin2d 4159	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ (𝑟 ∩ 𝐴) ∧ 𝑤 ∈ (𝑠 ∩ 𝐵)) → ⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵))
69	68	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))
70		inelcm 4419	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑢 ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵)) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)
71	67, 69, 70	syl2anc 585	. . . . . . . . . . . . . . . 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)) ≠ ∅)
72	71	ex 412	. . . . . . . . . . . . . . 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)) ≠ ∅))
73	72	exlimdvv 1936	. . . . . . . . . . . . . 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)) ≠ ∅))
74	60, 73	biimtrrid 243	. . . . . . . . . . . . 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)) ≠ ∅))
75	50, 59, 74	mp2and 700	. . . . . . . . . . . 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)) ≠ ∅)
76	75	expr 456	. . . . . . . . . . 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)) ≠ ∅))
77	76	rexlimdvva 3195	. . . . . . . . . 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)) ≠ ∅))
78	38, 77	syl5 34	. . . . . . . . 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)) ≠ ∅))
79	78	expd 415	. . . . . . . 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)) ≠ ∅)))
80	34, 79	sylbid 240	. . . . . . 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)) ≠ ∅)))
81	80	ralrimiv 3129	. . . . . 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)) ≠ ∅))
82		txtopon 23552	. . . . . . . . 9 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
83	82	ad2antrr 727	. . . . . . . 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)))
84		topontop 22874	. . . . . . . 8 ⊢ ((𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑅 ×_t 𝑆) ∈ Top)
85	83, 84	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)
86		xpss12 5649	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
87	86	ad2antlr 728	. . . . . . . 8 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
88		toponuni 22875	. . . . . . . . 9 ⊢ ((𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = ∪ (𝑅 ×_t 𝑆))
89	83, 88	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))
90	87, 89	sseqtrd 3972	. . . . . . 7 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝐴 × 𝐵) ⊆ ∪ (𝑅 ×_t 𝑆))
91	7	clsss3 23020	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ⊆ ∪ 𝑅) → ((cls‘𝑅)‘𝐴) ⊆ ∪ 𝑅)
92	2, 6, 91	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘𝑅)‘𝐴) ⊆ ∪ 𝑅)
93	92, 5	sseqtrrd 3973	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘𝑅)‘𝐴) ⊆ 𝑋)
94	93	sselda 3935	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑥 ∈ ((cls‘𝑅)‘𝐴)) → 𝑥 ∈ 𝑋)
95	94	adantrr 718	. . . . . . . . 9 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → 𝑥 ∈ 𝑋)
96	16	clsss3 23020	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Top ∧ 𝐵 ⊆ ∪ 𝑆) → ((cls‘𝑆)‘𝐵) ⊆ ∪ 𝑆)
97	11, 15, 96	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘𝑆)‘𝐵) ⊆ ∪ 𝑆)
98	97, 14	sseqtrrd 3973	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘𝑆)‘𝐵) ⊆ 𝑌)
99	98	sselda 3935	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵)) → 𝑦 ∈ 𝑌)
100	99	adantrl 717	. . . . . . . . 9 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → 𝑦 ∈ 𝑌)
101	95, 100	opelxpd 5673	. . . . . . . 8 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
102	101, 89	eleqtrd 2839	. . . . . . 7 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ⟨𝑥, 𝑦⟩ ∈ ∪ (𝑅 ×_t 𝑆))
103	27	elcls 23034	. . . . . . 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)) ≠ ∅)))
104	85, 90, 102, 103	syl3anc 1374	. . . . . 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)) ≠ ∅)))
105	81, 104	mpbird 257	. . . . 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)))
106	105	ex 412	. . . 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))))
107	32, 106	biimtrid 242	. . 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))))
108	31, 107	relssdv 5747	. 2 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ⊆ ((cls‘(𝑅 ×_t 𝑆))‘(𝐴 × 𝐵)))
109	29, 108	eqssd 3953	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 206 ∧ wa 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 ⟨cop 4588 ∪ cuni 4865 × cxp 5632 Rel wrel 5639 ‘cfv 6502 (class class class)co 7370 Topctop 22854 TopOnctopon 22871 Clsdccld 22977 clsccl 22979 ×_t ctx 23521

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5529 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-ov 7373 df-oprab 7374 df-mpo 7375 df-1st 7945 df-2nd 7946 df-topgen 17377 df-top 22855 df-topon 22872 df-bases 22907 df-cld 22980 df-ntr 22981 df-cls 22982 df-tx 23523

This theorem is referenced by: clssubg 24070

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