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Theorem txcls 21326
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 20646 . . . . . 6 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
21ad2antrr 761 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝑅 ∈ Top)
3 simprl 793 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐴𝑋)
4 toponuni 20647 . . . . . . 7 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
54ad2antrr 761 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝑋 = 𝑅)
63, 5sseqtrd 3625 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐴 𝑅)
7 eqid 2621 . . . . . 6 𝑅 = 𝑅
87clscld 20770 . . . . 5 ((𝑅 ∈ Top ∧ 𝐴 𝑅) → ((cls‘𝑅)‘𝐴) ∈ (Clsd‘𝑅))
92, 6, 8syl2anc 692 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘𝑅)‘𝐴) ∈ (Clsd‘𝑅))
10 topontop 20646 . . . . . 6 (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
1110ad2antlr 762 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝑆 ∈ Top)
12 simprr 795 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
13 toponuni 20647 . . . . . . 7 (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = 𝑆)
1413ad2antlr 762 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝑌 = 𝑆)
1512, 14sseqtrd 3625 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐵 𝑆)
16 eqid 2621 . . . . . 6 𝑆 = 𝑆
1716clscld 20770 . . . . 5 ((𝑆 ∈ Top ∧ 𝐵 𝑆) → ((cls‘𝑆)‘𝐵) ∈ (Clsd‘𝑆))
1811, 15, 17syl2anc 692 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘𝑆)‘𝐵) ∈ (Clsd‘𝑆))
19 txcld 21325 . . . 4 ((((cls‘𝑅)‘𝐴) ∈ (Clsd‘𝑅) ∧ ((cls‘𝑆)‘𝐵) ∈ (Clsd‘𝑆)) → (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ∈ (Clsd‘(𝑅 ×t 𝑆)))
209, 18, 19syl2anc 692 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ∈ (Clsd‘(𝑅 ×t 𝑆)))
217sscls 20779 . . . . 5 ((𝑅 ∈ Top ∧ 𝐴 𝑅) → 𝐴 ⊆ ((cls‘𝑅)‘𝐴))
222, 6, 21syl2anc 692 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐴 ⊆ ((cls‘𝑅)‘𝐴))
2316sscls 20779 . . . . 5 ((𝑆 ∈ Top ∧ 𝐵 𝑆) → 𝐵 ⊆ ((cls‘𝑆)‘𝐵))
2411, 15, 23syl2anc 692 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐵 ⊆ ((cls‘𝑆)‘𝐵))
25 xpss12 5191 . . . 4 ((𝐴 ⊆ ((cls‘𝑅)‘𝐴) ∧ 𝐵 ⊆ ((cls‘𝑆)‘𝐵)) → (𝐴 × 𝐵) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
2622, 24, 25syl2anc 692 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → (𝐴 × 𝐵) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
27 eqid 2621 . . . 4 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
2827clsss2 20795 . . 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 692 . 2 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘(𝑅 ×t 𝑆))‘(𝐴 × 𝐵)) ⊆ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
30 relxp 5193 . . . 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 5111 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ↔ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵)))
33 eltx 21290 . . . . . . . . 9 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑢 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑧𝑢𝑟𝑅𝑠𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢)))
3433ad2antrr 761 . . . . . . . 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 2686 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝑟 × 𝑠) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠)))
3635anbi1d 740 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢)))
37362rexbidv 3051 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (∃𝑟𝑅𝑠𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) ↔ ∃𝑟𝑅𝑠𝑆 (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢)))
3837rspccva 3297 . . . . . . . . . 10 ((∀𝑧𝑢𝑟𝑅𝑠𝑆 (𝑧 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑢) → ∃𝑟𝑅𝑠𝑆 (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑢))
392ad2antrr 761 . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . . 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 801 . . . . . . . . . . . . . . 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 803 . . . . . . . . . . . . . . . . 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 5111 . . . . . . . . . . . . . . . . 17 (⟨𝑥, 𝑦⟩ ∈ (𝑟 × 𝑠) ↔ (𝑥𝑟𝑦𝑠))
4543, 44sylib 208 . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . 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 20798 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Top ∧ 𝐴 𝑅𝑥 ∈ ((cls‘𝑅)‘𝐴)) ∧ (𝑟𝑅𝑥𝑟)) → (𝑟𝐴) ≠ ∅)
4839, 40, 41, 42, 46, 47syl32anc 1331 . . . . . . . . . . . . . 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 3912 . . . . . . . . . . . . . 14 ((𝑟𝐴) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑟𝐴))
5048, 49sylib 208 . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . . 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 802 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . 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 20798 . . . . . . . . . . . . . . 15 (((𝑆 ∈ Top ∧ 𝐵 𝑆𝑦 ∈ ((cls‘𝑆)‘𝐵)) ∧ (𝑠𝑆𝑦𝑠)) → (𝑠𝐵) ≠ ∅)
5751, 52, 53, 54, 55, 56syl32anc 1331 . . . . . . . . . . . . . 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 3912 . . . . . . . . . . . . . 14 ((𝑠𝐵) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑠𝐵))
5957, 58sylib 208 . . . . . . . . . . . . 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 eeanv 2181 . . . . . . . . . . . . . 14 (∃𝑧𝑤(𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵)) ↔ (∃𝑧 𝑧 ∈ (𝑟𝐴) ∧ ∃𝑤 𝑤 ∈ (𝑠𝐵)))
61 inss1 3816 . . . . . . . . . . . . . . . . . . 19 ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) ⊆ (𝑟 × 𝑠)
62 opelxpi 5113 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵)) → ⟨𝑧, 𝑤⟩ ∈ ((𝑟𝐴) × (𝑠𝐵)))
63 inxp 5219 . . . . . . . . . . . . . . . . . . . 20 ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟𝐴) × (𝑠𝐵))
6462, 63syl6eleqr 2709 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵)) → ⟨𝑧, 𝑤⟩ ∈ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
6561, 64sseldi 3585 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵)) → ⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠))
66 simprrr 804 . . . . . . . . . . . . . . . . . . 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 3587 . . . . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . . . . 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 3817 . . . . . . . . . . . . . . . . . . 19 ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
7069, 64sseldi 3585 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ (𝑟𝐴) ∧ 𝑤 ∈ (𝑠𝐵)) → ⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵))
7170adantl 482 . . . . . . . . . . . . . . . . 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 4009 . . . . . . . . . . . . . . . . 17 ((⟨𝑧, 𝑤⟩ ∈ 𝑢 ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐴 × 𝐵)) → (𝑢 ∩ (𝐴 × 𝐵)) ≠ ∅)
7368, 71, 72syl2anc 692 . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . 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 1859 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . 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 714 . . . . . . . . . . . 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 642 . . . . . . . . . . 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 3032 . . . . . . . . . 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 452 . . . . . . . 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 230 . . . . . . 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 2960 . . . . . 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 21313 . . . . . . . . 9 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
8584ad2antrr 761 . . . . . . . 8 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
86 topontop 20646 . . . . . . . 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 5191 . . . . . . . . 9 ((𝐴𝑋𝐵𝑌) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
8988ad2antlr 762 . . . . . . . 8 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
90 toponuni 20647 . . . . . . . . 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 3625 . . . . . . 7 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → (𝐴 × 𝐵) ⊆ (𝑅 ×t 𝑆))
937clsss3 20782 . . . . . . . . . . . . 13 ((𝑅 ∈ Top ∧ 𝐴 𝑅) → ((cls‘𝑅)‘𝐴) ⊆ 𝑅)
942, 6, 93syl2anc 692 . . . . . . . . . . . 12 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘𝑅)‘𝐴) ⊆ 𝑅)
9594, 5sseqtr4d 3626 . . . . . . . . . . 11 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘𝑅)‘𝐴) ⊆ 𝑋)
9695sselda 3587 . . . . . . . . . 10 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑥 ∈ ((cls‘𝑅)‘𝐴)) → 𝑥𝑋)
9796adantrr 752 . . . . . . . . 9 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → 𝑥𝑋)
9816clsss3 20782 . . . . . . . . . . . . 13 ((𝑆 ∈ Top ∧ 𝐵 𝑆) → ((cls‘𝑆)‘𝐵) ⊆ 𝑆)
9911, 15, 98syl2anc 692 . . . . . . . . . . . 12 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘𝑆)‘𝐵) ⊆ 𝑆)
10099, 14sseqtr4d 3626 . . . . . . . . . . 11 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘𝑆)‘𝐵) ⊆ 𝑌)
101100sselda 3587 . . . . . . . . . 10 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵)) → 𝑦𝑌)
102101adantrl 751 . . . . . . . . 9 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → 𝑦𝑌)
103 opelxpi 5113 . . . . . . . . 9 ((𝑥𝑋𝑦𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
10497, 102, 103syl2anc 692 . . . . . . . 8 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
105104, 91eleqtrd 2700 . . . . . . 7 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) ∧ (𝑥 ∈ ((cls‘𝑅)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝑆)‘𝐵))) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ×t 𝑆))
10627elcls 20796 . . . . . . 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 1323 . . . . . 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 247 . . . . 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 450 . . . 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 232 . . 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 5178 . 2 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)) ⊆ ((cls‘(𝑅 ×t 𝑆))‘(𝐴 × 𝐵)))
11229, 111eqssd 3604 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 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  cin 3558  wss 3559  c0 3896  cop 4159   cuni 4407   × cxp 5077  Rel wrel 5084  cfv 5852  (class class class)co 6610  Topctop 20626  TopOnctopon 20643  Clsdccld 20739  clsccl 20741   ×t ctx 21282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-topgen 16032  df-top 20627  df-topon 20644  df-bases 20670  df-cld 20742  df-ntr 20743  df-cls 20744  df-tx 21284
This theorem is referenced by:  clssubg  21831
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