Theorem txconn 23676

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 23404	. . 3 ⊢ (𝑅 ∈ Conn → 𝑅 ∈ Top)
2		conntop 23404	. . 3 ⊢ (𝑆 ∈ Conn → 𝑆 ∈ Top)
3		txtop 23556	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 603	. 2 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑅 ×t 𝑆) ∈ Top)
5		neq0 4283	. . . . . . 7 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑧 𝑧 ∈ 𝑥)
6		simplr 775	. . . . . . . . . . . 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 4136	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ (𝑅 ×t 𝑆))
8		elssuni 4872	. . . . . . . . . . 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 779	. . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑅 = ∪ 𝑅
16		eqid 2741	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑆 = ∪ 𝑆
17	15, 16	txuni 23579	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
18	12, 14, 17	syl2anc 591	. . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑤 ∈ (∪ 𝑅 × ∪ 𝑆))
20		1st2nd2 7974	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (∪ 𝑅 × ∪ 𝑆) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘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 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
22		xp2nd 7968	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (∪ 𝑅 × ∪ 𝑆) → (2nd ‘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))) → (2nd ‘𝑤) ∈ ∪ 𝑆)
24		eqid 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) = (𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩)
25	24	mptpreima 6193	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) “ 𝑥) = {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥}
26		toptopon2 22905	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
27	14, 26	sylib 220	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑆 ∈ (TopOn‘∪ 𝑆))
28		toptopon2 22905	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
29	12, 28	sylib 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑅 ∈ (TopOn‘∪ 𝑅))
30		xp1st 7967	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ (∪ 𝑅 × ∪ 𝑆) → (1st ‘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))) → (1st ‘𝑤) ∈ ∪ 𝑅)
32	27, 29, 31	cnmptc 23649	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑆 ↦ (1st ‘𝑤)) ∈ (𝑆 Cn 𝑅))
33	27	cnmptid 23648	. . . . . . . . . . . . . . . . . . 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 23652	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) ∈ (𝑆 Cn (𝑅 ×t 𝑆)))
35		simplr 775	. . . . . . . . . . . . . . . . . . 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 4136	. . . . . . . . . . . . . . . . . 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 23252	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) ∈ (𝑆 Cn (𝑅 ×t 𝑆)) ∧ 𝑥 ∈ (𝑅 ×t 𝑆)) → (◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) “ 𝑥) ∈ 𝑆)
38	34, 36, 37	syl2anc 591	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) “ 𝑥) ∈ 𝑆)
39	25, 38	eqeltrrid 2846	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ∈ 𝑆)
40		simprl 777	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑧 ∈ 𝑥)
41		elunii 4846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑥 ∈ (𝑅 ×t 𝑆)) → 𝑧 ∈ ∪ (𝑅 ×t 𝑆))
42	40, 36, 41	syl2anc 591	. . . . . . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑧 ∈ (∪ 𝑅 × ∪ 𝑆))
44		xp2nd 7968	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (∪ 𝑅 × ∪ 𝑆) → (2nd ‘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))) → (2nd ‘𝑧) ∈ ∪ 𝑆)
46		eqid 2741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) = (𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩)
47	46	mptpreima 6193	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) “ 𝑥) = {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥}
48	29	cnmptid 23648	. . . . . . . . . . . . . . . . . . . . . . . 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 23649	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑅 ↦ (2nd ‘𝑧)) ∈ (𝑅 Cn 𝑆))
50	29, 48, 49	cnmpt1t 23652	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) ∈ (𝑅 Cn (𝑅 ×t 𝑆)))
51		cnima 23252	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) ∈ (𝑅 Cn (𝑅 ×t 𝑆)) ∧ 𝑥 ∈ (𝑅 ×t 𝑆)) → (◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) “ 𝑥) ∈ 𝑅)
52	50, 36, 51	syl2anc 591	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) “ 𝑥) ∈ 𝑅)
53	47, 52	eqeltrrid 2846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} ∈ 𝑅)
54		xp1st 7967	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ (∪ 𝑅 × ∪ 𝑆) → (1st ‘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))) → (1st ‘𝑧) ∈ ∪ 𝑅)
56		1st2nd2 7974	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ (∪ 𝑅 × ∪ 𝑆) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘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 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
58	57, 40	eqeltrrd 2842	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ 𝑥)
59		opeq1 4807	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (1st ‘𝑧) → ⟨𝑎, (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
60	59	eleq1d 2826	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (1st ‘𝑧) → (⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥 ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ 𝑥))
61	60	rspcev 3562	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1st ‘𝑧) ∈ ∪ 𝑅 ∧ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ 𝑥) → ∃𝑎 ∈ ∪ 𝑅⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥)
62	55, 58, 61	syl2anc 591	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → ∃𝑎 ∈ ∪ 𝑅⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥)
63		rabn0 4320	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} ≠ ∅ ↔ ∃𝑎 ∈ ∪ 𝑅⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥)
64	62, 63	sylibr 236	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} ≠ ∅)
65	35	elin2d 4137	. . . . . . . . . . . . . . . . . . . . . . 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 23255	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) ∈ (𝑅 Cn (𝑅 ×t 𝑆)) ∧ 𝑥 ∈ (Clsd‘(𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) “ 𝑥) ∈ (Clsd‘𝑅))
67	50, 65, 66	syl2anc 591	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) “ 𝑥) ∈ (Clsd‘𝑅))
68	47, 67	eqeltrrid 2846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} ∈ (Clsd‘𝑅))
69	15, 11, 53, 64, 68	connclo 23402	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} = ∪ 𝑅)
70	31, 69	eleqtrrd 2844	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (1st ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥})
71		opeq1 4807	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (1st ‘𝑤) → ⟨𝑎, (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩)
72	71	eleq1d 2826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (1st ‘𝑤) → (⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥))
73	72	elrab 3631	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} ↔ ((1st ‘𝑤) ∈ ∪ 𝑅 ∧ ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥))
74	73	simprbi 499	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} → ⟨(1st ‘𝑤), (2nd ‘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))) → ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥)
76		opeq2 4808	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (2nd ‘𝑧) → ⟨(1st ‘𝑤), 𝑎⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩)
77	76	eleq1d 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (2nd ‘𝑧) → (⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥))
78	77	rspcev 3562	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘𝑧) ∈ ∪ 𝑆 ∧ ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥) → ∃𝑎 ∈ ∪ 𝑆⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥)
79	45, 75, 78	syl2anc 591	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → ∃𝑎 ∈ ∪ 𝑆⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥)
80		rabn0 4320	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ≠ ∅ ↔ ∃𝑎 ∈ ∪ 𝑆⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥)
81	79, 80	sylibr 236	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ≠ ∅)
82		cnclima 23255	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) ∈ (𝑆 Cn (𝑅 ×t 𝑆)) ∧ 𝑥 ∈ (Clsd‘(𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) “ 𝑥) ∈ (Clsd‘𝑆))
83	34, 65, 82	syl2anc 591	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) “ 𝑥) ∈ (Clsd‘𝑆))
84	25, 83	eqeltrrid 2846	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ∈ (Clsd‘𝑆))
85	16, 13, 39, 81, 84	connclo 23402	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} = ∪ 𝑆)
86	23, 85	eleqtrrd 2844	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (2nd ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥})
87		opeq2 4808	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (2nd ‘𝑤) → ⟨(1st ‘𝑤), 𝑎⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
88	87	eleq1d 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (2nd ‘𝑤) → (⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑥))
89	88	elrab 3631	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ↔ ((2nd ‘𝑤) ∈ ∪ 𝑆 ∧ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑥))
90	89	simprbi 499	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} → ⟨(1st ‘𝑤), (2nd ‘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))) → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑥)
92	21, 91	eqeltrd 2841	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑤 ∈ 𝑥)
93	92	expr 458	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → (𝑤 ∈ ∪ (𝑅 ×t 𝑆) → 𝑤 ∈ 𝑥))
94	93	ssrdv 3923	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → ∪ (𝑅 ×t 𝑆) ⊆ 𝑥)
95	9, 94	eqssd 3934	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 = ∪ (𝑅 ×t 𝑆))
96	95	ex 414	. . . . . . . 8 ⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (𝑧 ∈ 𝑥 → 𝑥 = ∪ (𝑅 ×t 𝑆)))
97	96	exlimdv 1941	. . . . . . 7 ⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (∃𝑧 𝑧 ∈ 𝑥 → 𝑥 = ∪ (𝑅 ×t 𝑆)))
98	5, 97	biimtrid 244	. . . . . 6 ⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (¬ 𝑥 = ∅ → 𝑥 = ∪ (𝑅 ×t 𝑆)))
99	98	orrd 870	. . . . 5 ⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×t 𝑆)))
100	99	ex 414	. . . 4 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) → (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×t 𝑆))))
101		vex 3437	. . . . 5 ⊢ 𝑥 ∈ V
102	101	elpr 4583	. . . 4 ⊢ (𝑥 ∈ {∅, ∪ (𝑅 ×t 𝑆)} ↔ (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×t 𝑆)))
103	100, 102	imbitrrdi 254	. . 3 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) → 𝑥 ∈ {∅, ∪ (𝑅 ×t 𝑆)}))
104	103	ssrdv 3923	. 2 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) ⊆ {∅, ∪ (𝑅 ×t 𝑆)})
105		eqid 2741	. . 3 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
106	105	isconn2 23401	. 2 ⊢ ((𝑅 ×t 𝑆) ∈ Conn ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) ⊆ {∅, ∪ (𝑅 ×t 𝑆)}))
107	4, 104, 106	sylanbrc 590	1 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑅 ×t 𝑆) ∈ Conn)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ wo 854   = wceq 1548  ∃wex 1787   ∈ wcel 2121   ≠ wne 2936  ∃wrex 3065  {crab 3393   ∩ cin 3884   ⊆ wss 3885  ∅c0 4264  {cpr 4560  ⟨cop 4564  ∪ cuni 4841   ↦ cmpt 5156   × cxp 5619  ^◡ccnv 5620   “ cima 5624  ‘cfv 6489  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Topctop 22880  TopOnctopon 22897  Clsdccld 23003   Cn ccn 23211  Conncconn 23398   ×_t ctx 23547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8769  df-topgen 17401  df-top 22881  df-topon 22898  df-bases 22933  df-cld 23006  df-cn 23214  df-cnp 23215  df-conn 23399  df-tx 23549

This theorem is referenced by:  cvmlift2lem9  35554  cvmlift2lem13  35558