Theorem txconn 22748

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 22476	. . 3 ⊢ (𝑅 ∈ Conn → 𝑅 ∈ Top)
2		conntop 22476	. . 3 ⊢ (𝑆 ∈ Conn → 𝑆 ∈ Top)
3		txtop 22628	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 595	. 2 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑅 ×t 𝑆) ∈ Top)
5		neq0 4276	. . . . . . 7 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑧 𝑧 ∈ 𝑥)
6		simplr 765	. . . . . . . . . . . 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 4128	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ (𝑅 ×t 𝑆))
8		elssuni 4868	. . . . . . . . . . 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 769	. . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑅 = ∪ 𝑅
16		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑆 = ∪ 𝑆
17	15, 16	txuni 22651	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
18	12, 14, 17	syl2anc 583	. . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑤 ∈ (∪ 𝑅 × ∪ 𝑆))
20		1st2nd2 7843	. . . . . . . . . . . . . 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 7837	. . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) = (𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩)
25	24	mptpreima 6130	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) “ 𝑥) = {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥}
26		toptopon2 21975	. . . . . . . . . . . . . . . . . . . 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 21975	. . . . . . . . . . . . . . . . . . . . 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 7836	. . . . . . . . . . . . . . . . . . . . 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 22721	. . . . . . . . . . . . . . . . . . 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 22720	. . . . . . . . . . . . . . . . . . 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 22724	. . . . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . 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 4128	. . . . . . . . . . . . . . . . . 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 22324	. . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑧 ∈ 𝑥)
41		elunii 4841	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑥 ∈ (𝑅 ×t 𝑆)) → 𝑧 ∈ ∪ (𝑅 ×t 𝑆))
42	40, 36, 41	syl2anc 583	. . . . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑧 ∈ (∪ 𝑅 × ∪ 𝑆))
44		xp2nd 7837	. . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) = (𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩)
47	46	mptpreima 6130	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) “ 𝑥) = {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥}
48	29	cnmptid 22720	. . . . . . . . . . . . . . . . . . . . . . . 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 22721	. . . . . . . . . . . . . . . . . . . . . . . 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 22724	. . . . . . . . . . . . . . . . . . . . . . 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 22324	. . . . . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . . . . . . . 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 7836	. . . . . . . . . . . . . . . . . . . . . . . 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 7843	. . . . . . . . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . . . . . . 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 4801	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (1st ‘𝑧) → ⟨𝑎, (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
60	59	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (1st ‘𝑧) → (⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥 ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ 𝑥))
61	60	rspcev 3552	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1st ‘𝑧) ∈ ∪ 𝑅 ∧ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ 𝑥) → ∃𝑎 ∈ ∪ 𝑅⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥)
62	55, 58, 61	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . 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 4316	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} ≠ ∅ ↔ ∃𝑎 ∈ ∪ 𝑅⟨𝑎, (2nd ‘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, (2nd ‘𝑧)⟩ ∈ 𝑥} ≠ ∅)
65	35	elin2d 4129	. . . . . . . . . . . . . . . . . . . . . . 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 22327	. . . . . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . . . . . . . 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 22474	. . . . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . . . . . 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 4801	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (1st ‘𝑤) → ⟨𝑎, (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩)
72	71	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (1st ‘𝑤) → (⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥))
73	72	elrab 3617	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} ↔ ((1st ‘𝑤) ∈ ∪ 𝑅 ∧ ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥))
74	73	simprbi 496	. . . . . . . . . . . . . . . . . . 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 4802	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (2nd ‘𝑧) → ⟨(1st ‘𝑤), 𝑎⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩)
77	76	eleq1d 2823	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (2nd ‘𝑧) → (⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥))
78	77	rspcev 3552	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘𝑧) ∈ ∪ 𝑆 ∧ ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥) → ∃𝑎 ∈ ∪ 𝑆⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥)
79	45, 75, 78	syl2anc 583	. . . . . . . . . . . . . . . . 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 4316	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ≠ ∅ ↔ ∃𝑎 ∈ ∪ 𝑆⟨(1st ‘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 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ≠ ∅)
82		cnclima 22327	. . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . . 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 22474	. . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . 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 4802	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (2nd ‘𝑤) → ⟨(1st ‘𝑤), 𝑎⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
88	87	eleq1d 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (2nd ‘𝑤) → (⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑥))
89	88	elrab 3617	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ↔ ((2nd ‘𝑤) ∈ ∪ 𝑆 ∧ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑥))
90	89	simprbi 496	. . . . . . . . . . . . . 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 2839	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑤 ∈ 𝑥)
93	92	expr 456	. . . . . . . . . . 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 412	. . . . . . . 8 ⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (𝑧 ∈ 𝑥 → 𝑥 = ∪ (𝑅 ×t 𝑆)))
97	96	exlimdv 1937	. . . . . . 7 ⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (∃𝑧 𝑧 ∈ 𝑥 → 𝑥 = ∪ (𝑅 ×t 𝑆)))
98	5, 97	syl5bi 241	. . . . . 6 ⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (¬ 𝑥 = ∅ → 𝑥 = ∪ (𝑅 ×t 𝑆)))
99	98	orrd 859	. . . . 5 ⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×t 𝑆)))
100	99	ex 412	. . . 4 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) → (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×t 𝑆))))
101		vex 3426	. . . . 5 ⊢ 𝑥 ∈ V
102	101	elpr 4581	. . . 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 3923	. 2 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) ⊆ {∅, ∪ (𝑅 ×t 𝑆)})
105		eqid 2738	. . 3 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
106	105	isconn2 22473	. 2 ⊢ ((𝑅 ×t 𝑆) ∈ Conn ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) ⊆ {∅, ∪ (𝑅 ×t 𝑆)}))
107	4, 104, 106	sylanbrc 582	1 ⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑅 ×t 𝑆) ∈ Conn)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064  {crab 3067   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {cpr 4560  ⟨cop 4564  ∪ cuni 4836   ↦ cmpt 5153   × cxp 5578  ^◡ccnv 5579   “ cima 5583  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  Topctop 21950  TopOnctopon 21967  Clsdccld 22075   Cn ccn 22283  Conncconn 22470   ×_t ctx 22619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-topgen 17071  df-top 21951  df-topon 21968  df-bases 22004  df-cld 22078  df-cn 22286  df-cnp 22287  df-conn 22471  df-tx 22621

This theorem is referenced by:  cvmlift2lem9  33173  cvmlift2lem13  33177