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Theorem txcls 21778
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)) ∧ (𝐴𝑋𝐵𝑌)) → ((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.
StepHypRef Expression
1 topontop 21088 . . . . . 6 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
21ad2antrr 717 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝑅 ∈ Top)
3 simprl 787 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐴𝑋)
4 toponuni 21089 . . . . . . 7 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
54ad2antrr 717 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝑋 = 𝑅)
63, 5sseqtrd 3866 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐴 𝑅)
7 eqid 2825 . . . . . 6 𝑅 = 𝑅
87clscld 21222 . . . . 5 ((𝑅 ∈ Top ∧ 𝐴 𝑅) → ((cls‘𝑅)‘𝐴) ∈ (Clsd‘𝑅))
92, 6, 8syl2anc 579 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘𝑅)‘𝐴) ∈ (Clsd‘𝑅))
10 topontop 21088 . . . . . 6 (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
1110ad2antlr 718 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝑆 ∈ Top)
12 simprr 789 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
13 toponuni 21089 . . . . . . 7 (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = 𝑆)
1413ad2antlr 718 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝑌 = 𝑆)
1512, 14sseqtrd 3866 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐵 𝑆)
16 eqid 2825 . . . . . 6 𝑆 = 𝑆
1716clscld 21222 . . . . 5 ((𝑆 ∈ Top ∧ 𝐵 𝑆) → ((cls‘𝑆)‘𝐵) ∈ (Clsd‘𝑆))
1811, 15, 17syl2anc 579 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘𝑆)‘𝐵) ∈ (Clsd‘𝑆))
19 txcld 21777 . . . 4 ((((cls‘𝑅)‘𝐴) ∈ (Clsd‘𝑅) ∧ ((cls‘𝑆)‘𝐵) ∈ (Clsd‘𝑆)) → (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ∈ (Clsd‘(𝑅 ×t 𝑆)))
209, 18, 19syl2anc 579 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ∈ (Clsd‘(𝑅 ×t 𝑆)))
217sscls 21231 . . . . 5 ((𝑅 ∈ Top ∧ 𝐴 𝑅) → 𝐴 ⊆ ((cls‘𝑅)‘𝐴))
222, 6, 21syl2anc 579 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐴 ⊆ ((cls‘𝑅)‘𝐴))
2316sscls 21231 . . . . 5 ((𝑆 ∈ Top ∧ 𝐵 𝑆) → 𝐵 ⊆ ((cls‘𝑆)‘𝐵))
2411, 15, 23syl2anc 579 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐵 ⊆ ((cls‘𝑆)‘𝐵))
25 xpss12 5357 . . . 4 ((𝐴 ⊆ ((cls‘𝑅)‘𝐴) ∧ 𝐵 ⊆ ((cls‘𝑆)‘𝐵)) → (𝐴 × 𝐵) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
2622, 24, 25syl2anc 579 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → (𝐴 × 𝐵) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
27 eqid 2825 . . . 4 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
2827clsss2 21247 . . 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)))
2920, 26, 28syl2anc 579 . 2 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘(𝑅 ×t 𝑆))‘(𝐴 × 𝐵)) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
30 relxp 5360 . . . 4 Rel (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵))
3130a1i 11 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → Rel (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
32 opelxp 5378 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ↔ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵)))
33 eltx 21742 . . . . . . . . 9 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑢 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑧𝑢𝑟𝑅𝑠𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢)))
3433ad2antrr 717 . . . . . . . 8 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝑢 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑧𝑢𝑟𝑅𝑠𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢)))
35 eleq1 2894 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝑟 × 𝑠) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠)))
3635anbi1d 623 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢)))
37362rexbidv 3267 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (∃𝑟𝑅𝑠𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) ↔ ∃𝑟𝑅𝑠𝑆 (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢)))
3837rspccva 3525 . . . . . . . . . 10 ((∀𝑧𝑢𝑟𝑅𝑠𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑢) → ∃𝑟𝑅𝑠𝑆 (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))
392ad2antrr 717 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑅 ∈ Top)
406ad2antrr 717 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝐴 𝑅)
41 simplrl 795 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑥 ∈ ((cls‘𝑅)‘𝐴))
42 simprll 797 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑟𝑅)
43 simprrl 799 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → ⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠))
44 opelxp 5378 . . . . . . . . . . . . . . . . 17 (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ↔ (𝑥𝑟𝑦𝑠))
4543, 44sylib 210 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → (𝑥𝑟𝑦𝑠))
4645simpld 490 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑥𝑟)
477clsndisj 21250 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Top ∧ 𝐴 𝑅𝑥 ∈ ((cls‘𝑅)‘𝐴)) ∧ (𝑟𝑅𝑥𝑟)) → (𝑟𝐴) ≠ ∅)
4839, 40, 41, 42, 46, 47syl32anc 1501 . . . . . . . . . . . . . 14 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → (𝑟𝐴) ≠ ∅)
49 n0 4160 . . . . . . . . . . . . . 14 ((𝑟𝐴) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑟𝐴))
5048, 49sylib 210 . . . . . . . . . . . . 13 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → ∃𝑧 𝑧 ∈ (𝑟𝐴))
5111ad2antrr 717 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑆 ∈ Top)
5215ad2antrr 717 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝐵 𝑆)
53 simplrr 796 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑦 ∈ ((cls‘𝑆)‘𝐵))
54 simprlr 798 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑠𝑆)
5545simprd 491 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → 𝑦𝑠)
5616clsndisj 21250 . . . . . . . . . . . . . . 15 (((𝑆 ∈ Top ∧ 𝐵 𝑆𝑦 ∈ ((cls‘𝑆)‘𝐵)) ∧ (𝑠𝑆𝑦𝑠)) → (𝑠𝐵) ≠ ∅)
5751, 52, 53, 54, 55, 56syl32anc 1501 . . . . . . . . . . . . . 14 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → (𝑠𝐵) ≠ ∅)
58 n0 4160 . . . . . . . . . . . . . 14 ((𝑠𝐵) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑠𝐵))
5957, 58sylib 210 . . . . . . . . . . . . 13 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → ∃𝑤 𝑤 ∈ (𝑠𝐵))
60 exdistrv 2054 . . . . . . . . . . . . . 14 (∃𝑧𝑤(𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵)) ↔ (∃𝑧 𝑧 ∈ (𝑟𝐴) ∧ ∃𝑤 𝑤 ∈ (𝑠𝐵)))
61 inss1 4057 . . . . . . . . . . . . . . . . . . 19 ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) ⊆ (𝑟 × 𝑠)
62 opelxpi 5379 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵)) → ⟨𝑧, 𝑤⟩ ∈ ((𝑟𝐴) × (𝑠𝐵)))
63 inxp 5487 . . . . . . . . . . . . . . . . . . . 20 ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟𝐴) × (𝑠𝐵))
6462, 63syl6eleqr 2917 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵)) → ⟨𝑧, 𝑤⟩ ∈ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
6561, 64sseldi 3825 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵)) → ⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠))
66 simprrr 800 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → (𝑟 × 𝑠) ⊆ 𝑢)
6766sselda 3827 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠)) → ⟨𝑧, 𝑤⟩ ∈ 𝑢)
6865, 67sylan2 586 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) ∧ (𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵))) → ⟨𝑧, 𝑤⟩ ∈ 𝑢)
69 inss2 4058 . . . . . . . . . . . . . . . . . . 19 ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
7069, 64sseldi 3825 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵)) → ⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵))
7170adantl 475 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) ∧ (𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵))) → ⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵))
72 inelcm 4256 . . . . . . . . . . . . . . . . 17 ((⟨𝑧, 𝑤⟩ ∈ 𝑢 ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵)) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)
7368, 71, 72syl2anc 579 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) ∧ (𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵))) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)
7473ex 403 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → ((𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵)) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
7574exlimdvv 2033 . . . . . . . . . . . . . 14 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → (∃𝑧𝑤(𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵)) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
7660, 75syl5bir 235 . . . . . . . . . . . . 13 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → ((∃𝑧 𝑧 ∈ (𝑟𝐴) ∧ ∃𝑤 𝑤 ∈ (𝑠𝐵)) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
7750, 59, 76mp2and 690 . . . . . . . . . . . 12 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)
7877expr 450 . . . . . . . . . . 11 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) ∧ (𝑟𝑅𝑠𝑆)) → ((⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
7978rexlimdvva 3248 . . . . . . . . . 10 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (∃𝑟𝑅𝑠𝑆 (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
8038, 79syl5 34 . . . . . . . . 9 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ((∀𝑧𝑢𝑟𝑅𝑠𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑢) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
8180expd 406 . . . . . . . 8 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (∀𝑧𝑢𝑟𝑅𝑠𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) → (⟨𝑥, 𝑦⟩ ∈ 𝑢 → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)))
8234, 81sylbid 232 . . . . . . 7 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝑢 ∈ (𝑅 ×t 𝑆) → (⟨𝑥, 𝑦⟩ ∈ 𝑢 → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)))
8382ralrimiv 3174 . . . . . 6 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ∀𝑢 ∈ (𝑅 ×t 𝑆)(⟨𝑥, 𝑦⟩ ∈ 𝑢 → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅))
84 txtopon 21765 . . . . . . . . 9 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
8584ad2antrr 717 . . . . . . . 8 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
86 topontop 21088 . . . . . . . 8 ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑅 ×t 𝑆) ∈ Top)
8785, 86syl 17 . . . . . . 7 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝑅 ×t 𝑆) ∈ Top)
88 xpss12 5357 . . . . . . . . 9 ((𝐴𝑋𝐵𝑌) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
8988ad2antlr 718 . . . . . . . 8 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
90 toponuni 21089 . . . . . . . . 9 ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
9185, 90syl 17 . . . . . . . 8 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
9289, 91sseqtrd 3866 . . . . . . 7 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝐴 × 𝐵) ⊆ (𝑅 ×t 𝑆))
937clsss3 21234 . . . . . . . . . . . . 13 ((𝑅 ∈ Top ∧ 𝐴 𝑅) → ((cls‘𝑅)‘𝐴) ⊆ 𝑅)
942, 6, 93syl2anc 579 . . . . . . . . . . . 12 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘𝑅)‘𝐴) ⊆ 𝑅)
9594, 5sseqtr4d 3867 . . . . . . . . . . 11 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘𝑅)‘𝐴) ⊆ 𝑋)
9695sselda 3827 . . . . . . . . . 10 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑥 ∈ ((cls‘𝑅)‘𝐴)) → 𝑥𝑋)
9796adantrr 708 . . . . . . . . 9 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → 𝑥𝑋)
9816clsss3 21234 . . . . . . . . . . . . 13 ((𝑆 ∈ Top ∧ 𝐵 𝑆) → ((cls‘𝑆)‘𝐵) ⊆ 𝑆)
9911, 15, 98syl2anc 579 . . . . . . . . . . . 12 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘𝑆)‘𝐵) ⊆ 𝑆)
10099, 14sseqtr4d 3867 . . . . . . . . . . 11 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘𝑆)‘𝐵) ⊆ 𝑌)
101100sselda 3827 . . . . . . . . . 10 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵)) → 𝑦𝑌)
102101adantrl 707 . . . . . . . . 9 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → 𝑦𝑌)
103 opelxpi 5379 . . . . . . . . 9 ((𝑥𝑋𝑦𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
10497, 102, 103syl2anc 579 . . . . . . . 8 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
105104, 91eleqtrd 2908 . . . . . . 7 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ×t 𝑆))
10627elcls 21248 . . . . . . 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)) ≠ ∅)))
10787, 92, 105, 106syl3anc 1494 . . . . . 6 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (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)) ≠ ∅)))
10883, 107mpbird 249 . . . . 5 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ⟨𝑥, 𝑦⟩ ∈ ((cls‘(𝑅 ×t 𝑆))‘(𝐴 × 𝐵)))
109108ex 403 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵)) → ⟨𝑥, 𝑦⟩ ∈ ((cls‘(𝑅 ×t 𝑆))‘(𝐴 × 𝐵))))
11032, 109syl5bi 234 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → (⟨𝑥, 𝑦⟩ ∈ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) → ⟨𝑥, 𝑦⟩ ∈ ((cls‘(𝑅 ×t 𝑆))‘(𝐴 × 𝐵))))
11131, 110relssdv 5446 . 2 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ⊆ ((cls‘(𝑅 ×t 𝑆))‘(𝐴 × 𝐵)))
11229, 111eqssd 3844 1 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘(𝑅 ×t 𝑆))‘(𝐴 × 𝐵)) = (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wex 1878  wcel 2164  wne 2999  wral 3117  wrex 3118  cin 3797  wss 3798  c0 4144  cop 4403   cuni 4658   × cxp 5340  Rel wrel 5347  cfv 6123  (class class class)co 6905  Topctop 21068  TopOnctopon 21085  Clsdccld 21191  clsccl 21193   ×t ctx 21734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-topgen 16457  df-top 21069  df-topon 21086  df-bases 21121  df-cld 21194  df-ntr 21195  df-cls 21196  df-tx 21736
This theorem is referenced by:  clssubg  22282
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