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Theorem txconn 23192

Description: The topological product of two connected spaces is connected. (Contributed by Mario Carneiro, 29-Mar-2015.)

**Assertion**
Ref	Expression
txconn	⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑅 ×_t 𝑆) ∈ Conn)

Proof of Theorem txconn Dummy variables 𝑤 𝑎 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		conntop 22920	. . 3 ⊢ (𝑅 ∈ Conn → 𝑅 ∈ Top)
2		conntop 22920	. . 3 ⊢ (𝑆 ∈ Conn → 𝑆 ∈ Top)
3		txtop 23072	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×_t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑅 ×_t 𝑆) ∈ Top)
5		neq0 4345	. . . . . . 7 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑧 𝑧 ∈ 𝑥)
6		simplr 767	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆))))
7	6	elin1d 4198	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ (𝑅 ×_t 𝑆))
8		elssuni 4941	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑅 ×_t 𝑆) → 𝑥 ⊆ ∪ (𝑅 ×_t 𝑆))
9	7, 8	syl 17	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 ⊆ ∪ (𝑅 ×_t 𝑆))
10		simprr 771	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))
11		simplll 773	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑅 ∈ Conn)
12	11, 1	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑅 ∈ Top)
13		simpllr 774	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑆 ∈ Conn)
14	13, 2	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑆 ∈ Top)
15		eqid 2732	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑅 = ∪ 𝑅
16		eqid 2732	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑆 = ∪ 𝑆
17	15, 16	txuni 23095	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
18	12, 14, 17	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
19	10, 18	eleqtrrd 2836	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑤 ∈ (∪ 𝑅 × ∪ 𝑆))
20		1st2nd2 8013	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (∪ 𝑅 × ∪ 𝑆) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
21	19, 20	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
22		xp2nd 8007	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (∪ 𝑅 × ∪ 𝑆) → (2^nd ‘𝑤) ∈ ∪ 𝑆)
23	19, 22	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (2^nd ‘𝑤) ∈ ∪ 𝑆)
24		eqid 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ∪ 𝑆 ↦ ⟨(1^st ‘𝑤), 𝑎⟩) = (𝑎 ∈ ∪ 𝑆 ↦ ⟨(1^st ‘𝑤), 𝑎⟩)
25	24	mptpreima 6237	. . . . . . . . . . . . . . . . 17 ⊢ (^◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1^st ‘𝑤), 𝑎⟩) “ 𝑥) = {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥}
26		toptopon2 22419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
27	14, 26	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑆 ∈ (TopOn‘∪ 𝑆))
28		toptopon2 22419	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
29	12, 28	sylib 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑅 ∈ (TopOn‘∪ 𝑅))
30		xp1st 8006	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ (∪ 𝑅 × ∪ 𝑆) → (1^st ‘𝑤) ∈ ∪ 𝑅)
31	19, 30	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (1^st ‘𝑤) ∈ ∪ 𝑅)
32	27, 29, 31	cnmptc 23165	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (𝑎 ∈ ∪ 𝑆 ↦ (1^st ‘𝑤)) ∈ (𝑆 Cn 𝑅))
33	27	cnmptid 23164	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (𝑎 ∈ ∪ 𝑆 ↦ 𝑎) ∈ (𝑆 Cn 𝑆))
34	27, 32, 33	cnmpt1t 23168	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (𝑎 ∈ ∪ 𝑆 ↦ ⟨(1^st ‘𝑤), 𝑎⟩) ∈ (𝑆 Cn (𝑅 ×_t 𝑆)))
35		simplr 767	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆))))
36	35	elin1d 4198	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑥 ∈ (𝑅 ×_t 𝑆))
37		cnima 22768	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 ∈ ∪ 𝑆 ↦ ⟨(1^st ‘𝑤), 𝑎⟩) ∈ (𝑆 Cn (𝑅 ×_t 𝑆)) ∧ 𝑥 ∈ (𝑅 ×_t 𝑆)) → (^◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1^st ‘𝑤), 𝑎⟩) “ 𝑥) ∈ 𝑆)
38	34, 36, 37	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (^◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1^st ‘𝑤), 𝑎⟩) “ 𝑥) ∈ 𝑆)
39	25, 38	eqeltrrid 2838	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥} ∈ 𝑆)
40		simprl 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑧 ∈ 𝑥)
41		elunii 4913	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑥 ∈ (𝑅 ×_t 𝑆)) → 𝑧 ∈ ∪ (𝑅 ×_t 𝑆))
42	40, 36, 41	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑧 ∈ ∪ (𝑅 ×_t 𝑆))
43	42, 18	eleqtrrd 2836	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑧 ∈ (∪ 𝑅 × ∪ 𝑆))
44		xp2nd 8007	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (∪ 𝑅 × ∪ 𝑆) → (2^nd ‘𝑧) ∈ ∪ 𝑆)
45	43, 44	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (2^nd ‘𝑧) ∈ ∪ 𝑆)
46		eqid 2732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2^nd ‘𝑧)⟩) = (𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2^nd ‘𝑧)⟩)
47	46	mptpreima 6237	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (^◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2^nd ‘𝑧)⟩) “ 𝑥) = {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥}
48	29	cnmptid 23164	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (𝑎 ∈ ∪ 𝑅 ↦ 𝑎) ∈ (𝑅 Cn 𝑅))
49	29, 27, 45	cnmptc 23165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (𝑎 ∈ ∪ 𝑅 ↦ (2^nd ‘𝑧)) ∈ (𝑅 Cn 𝑆))
50	29, 48, 49	cnmpt1t 23168	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2^nd ‘𝑧)⟩) ∈ (𝑅 Cn (𝑅 ×_t 𝑆)))
51		cnima 22768	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2^nd ‘𝑧)⟩) ∈ (𝑅 Cn (𝑅 ×_t 𝑆)) ∧ 𝑥 ∈ (𝑅 ×_t 𝑆)) → (^◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2^nd ‘𝑧)⟩) “ 𝑥) ∈ 𝑅)
52	50, 36, 51	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (^◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2^nd ‘𝑧)⟩) “ 𝑥) ∈ 𝑅)
53	47, 52	eqeltrrid 2838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥} ∈ 𝑅)
54		xp1st 8006	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ (∪ 𝑅 × ∪ 𝑆) → (1^st ‘𝑧) ∈ ∪ 𝑅)
55	43, 54	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (1^st ‘𝑧) ∈ ∪ 𝑅)
56		1st2nd2 8013	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ (∪ 𝑅 × ∪ 𝑆) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
57	43, 56	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
58	57, 40	eqeltrrd 2834	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ 𝑥)
59		opeq1 4873	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (1^st ‘𝑧) → ⟨𝑎, (2^nd ‘𝑧)⟩ = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
60	59	eleq1d 2818	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (1^st ‘𝑧) → (⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥 ↔ ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ 𝑥))
61	60	rspcev 3612	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1^st ‘𝑧) ∈ ∪ 𝑅 ∧ ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ 𝑥) → ∃𝑎 ∈ ∪ 𝑅⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥)
62	55, 58, 61	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → ∃𝑎 ∈ ∪ 𝑅⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥)
63		rabn0 4385	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥} ≠ ∅ ↔ ∃𝑎 ∈ ∪ 𝑅⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥)
64	62, 63	sylibr 233	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥} ≠ ∅)
65	35	elin2d 4199	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑥 ∈ (Clsd‘(𝑅 ×_t 𝑆)))
66		cnclima 22771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2^nd ‘𝑧)⟩) ∈ (𝑅 Cn (𝑅 ×_t 𝑆)) ∧ 𝑥 ∈ (Clsd‘(𝑅 ×_t 𝑆))) → (^◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2^nd ‘𝑧)⟩) “ 𝑥) ∈ (Clsd‘𝑅))
67	50, 65, 66	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (^◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2^nd ‘𝑧)⟩) “ 𝑥) ∈ (Clsd‘𝑅))
68	47, 67	eqeltrrid 2838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥} ∈ (Clsd‘𝑅))
69	15, 11, 53, 64, 68	connclo 22918	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥} = ∪ 𝑅)
70	31, 69	eleqtrrd 2836	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (1^st ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥})
71		opeq1 4873	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (1^st ‘𝑤) → ⟨𝑎, (2^nd ‘𝑧)⟩ = ⟨(1^st ‘𝑤), (2^nd ‘𝑧)⟩)
72	71	eleq1d 2818	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (1^st ‘𝑤) → (⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥 ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑧)⟩ ∈ 𝑥))
73	72	elrab 3683	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1^st ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥} ↔ ((1^st ‘𝑤) ∈ ∪ 𝑅 ∧ ⟨(1^st ‘𝑤), (2^nd ‘𝑧)⟩ ∈ 𝑥))
74	73	simprbi 497	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1^st ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2^nd ‘𝑧)⟩ ∈ 𝑥} → ⟨(1^st ‘𝑤), (2^nd ‘𝑧)⟩ ∈ 𝑥)
75	70, 74	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → ⟨(1^st ‘𝑤), (2^nd ‘𝑧)⟩ ∈ 𝑥)
76		opeq2 4874	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (2^nd ‘𝑧) → ⟨(1^st ‘𝑤), 𝑎⟩ = ⟨(1^st ‘𝑤), (2^nd ‘𝑧)⟩)
77	76	eleq1d 2818	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (2^nd ‘𝑧) → (⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥 ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑧)⟩ ∈ 𝑥))
78	77	rspcev 3612	. . . . . . . . . . . . . . . . . 18 ⊢ (((2^nd ‘𝑧) ∈ ∪ 𝑆 ∧ ⟨(1^st ‘𝑤), (2^nd ‘𝑧)⟩ ∈ 𝑥) → ∃𝑎 ∈ ∪ 𝑆⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥)
79	45, 75, 78	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → ∃𝑎 ∈ ∪ 𝑆⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥)
80		rabn0 4385	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑎 ∈ ∪ 𝑆 ∣ ⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥} ≠ ∅ ↔ ∃𝑎 ∈ ∪ 𝑆⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥)
81	79, 80	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥} ≠ ∅)
82		cnclima 22771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 ∈ ∪ 𝑆 ↦ ⟨(1^st ‘𝑤), 𝑎⟩) ∈ (𝑆 Cn (𝑅 ×_t 𝑆)) ∧ 𝑥 ∈ (Clsd‘(𝑅 ×_t 𝑆))) → (^◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1^st ‘𝑤), 𝑎⟩) “ 𝑥) ∈ (Clsd‘𝑆))
83	34, 65, 82	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (^◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1^st ‘𝑤), 𝑎⟩) “ 𝑥) ∈ (Clsd‘𝑆))
84	25, 83	eqeltrrid 2838	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥} ∈ (Clsd‘𝑆))
85	16, 13, 39, 81, 84	connclo 22918	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥} = ∪ 𝑆)
86	23, 85	eleqtrrd 2836	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → (2^nd ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥})
87		opeq2 4874	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (2^nd ‘𝑤) → ⟨(1^st ‘𝑤), 𝑎⟩ = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
88	87	eleq1d 2818	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (2^nd ‘𝑤) → (⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥 ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ 𝑥))
89	88	elrab 3683	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥} ↔ ((2^nd ‘𝑤) ∈ ∪ 𝑆 ∧ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ 𝑥))
90	89	simprbi 497	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1^st ‘𝑤), 𝑎⟩ ∈ 𝑥} → ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ 𝑥)
91	86, 90	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ 𝑥)
92	21, 91	eqeltrd 2833	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×_t 𝑆))) → 𝑤 ∈ 𝑥)
93	92	expr 457	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → (𝑤 ∈ ∪ (𝑅 ×_t 𝑆) → 𝑤 ∈ 𝑥))
94	93	ssrdv 3988	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → ∪ (𝑅 ×_t 𝑆) ⊆ 𝑥)
95	9, 94	eqssd 3999	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 = ∪ (𝑅 ×_t 𝑆))
96	95	ex 413	. . . . . . . 8 ⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) → (𝑧 ∈ 𝑥 → 𝑥 = ∪ (𝑅 ×_t 𝑆)))
97	96	exlimdv 1936	. . . . . . 7 ⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) → (∃𝑧 𝑧 ∈ 𝑥 → 𝑥 = ∪ (𝑅 ×_t 𝑆)))
98	5, 97	biimtrid 241	. . . . . 6 ⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) → (¬ 𝑥 = ∅ → 𝑥 = ∪ (𝑅 ×_t 𝑆)))
99	98	orrd 861	. . . . 5 ⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆)))) → (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×_t 𝑆)))
100	99	ex 413	. . . 4 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆))) → (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×_t 𝑆))))
101		vex 3478	. . . . 5 ⊢ 𝑥 ∈ V
102	101	elpr 4651	. . . 4 ⊢ (𝑥 ∈ {∅, ∪ (𝑅 ×_t 𝑆)} ↔ (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×_t 𝑆)))
103	100, 102	syl6ibr 251	. . 3 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑥 ∈ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆))) → 𝑥 ∈ {∅, ∪ (𝑅 ×_t 𝑆)}))
104	103	ssrdv 3988	. 2 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆))) ⊆ {∅, ∪ (𝑅 ×_t 𝑆)})
105		eqid 2732	. . 3 ⊢ ∪ (𝑅 ×_t 𝑆) = ∪ (𝑅 ×_t 𝑆)
106	105	isconn2 22917	. 2 ⊢ ((𝑅 ×_t 𝑆) ∈ Conn ↔ ((𝑅 ×_t 𝑆) ∈ Top ∧ ((𝑅 ×_t 𝑆) ∩ (Clsd‘(𝑅 ×_t 𝑆))) ⊆ {∅, ∪ (𝑅 ×_t 𝑆)}))
107	4, 104, 106	sylanbrc 583	1 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑅 ×_t 𝑆) ∈ Conn)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 {crab 3432 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 {cpr 4630 ⟨cop 4634 ∪ cuni 4908 ↦ cmpt 5231 × cxp 5674 ^◡ccnv 5675 “ cima 5679 ‘cfv 6543 (class class class)co 7408 1^st c1st 7972 2^nd c2nd 7973 Topctop 22394 TopOnctopon 22411 Clsdccld 22519 Cn ccn 22727 Conncconn 22914 ×_t ctx 23063

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-map 8821 df-topgen 17388 df-top 22395 df-topon 22412 df-bases 22448 df-cld 22522 df-cn 22730 df-cnp 22731 df-conn 22915 df-tx 23065

This theorem is referenced by: cvmlift2lem9 34297 cvmlift2lem13 34301

W3C validator