Theorem txcon 21302

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

Assertion

Ref	Expression
txcon	⊢ ((𝑅 ∈ Con ∧ 𝑆 ∈ Con) → (𝑅 ×t 𝑆) ∈ Con)

Proof of Theorem txcon

Dummy variables 𝑤 𝑎 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		contop 21030	. . 3 ⊢ (𝑅 ∈ Con → 𝑅 ∈ Top)
2		contop 21030	. . 3 ⊢ (𝑆 ∈ Con → 𝑆 ∈ Top)
3		txtop 21182	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 493	. 2 ⊢ ((𝑅 ∈ Con ∧ 𝑆 ∈ Con) → (𝑅 ×t 𝑆) ∈ Top)
5		neq0 3889	. . . . . . 7 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑧 𝑧 ∈ 𝑥)
6		inss1 3795	. . . . . . . . . . . 12 ⊢ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) ⊆ (𝑅 ×t 𝑆)
7		simplr 788	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))))
8	6, 7	sseldi 3566	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ (𝑅 ×t 𝑆))
9		elssuni 4403	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑅 ×t 𝑆) → 𝑥 ⊆ ∪ (𝑅 ×t 𝑆))
10	8, 9	syl 17	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 ⊆ ∪ (𝑅 ×t 𝑆))
11		simprr 792	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑤 ∈ ∪ (𝑅 ×t 𝑆))
12		simplll 794	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑅 ∈ Con)
13	12, 1	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑅 ∈ Top)
14		simpllr 795	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑆 ∈ Con)
15	14, 2	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑆 ∈ Top)
16		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑅 = ∪ 𝑅
17		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑆 = ∪ 𝑆
18	16, 17	txuni 21205	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
19	13, 15, 18	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
20	11, 19	eleqtrrd 2691	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑤 ∈ (∪ 𝑅 × ∪ 𝑆))
21		1st2nd2 7096	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (∪ 𝑅 × ∪ 𝑆) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
22	20, 21	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
23		xp2nd 7090	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (∪ 𝑅 × ∪ 𝑆) → (2nd ‘𝑤) ∈ ∪ 𝑆)
24	20, 23	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (2nd ‘𝑤) ∈ ∪ 𝑆)
25		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) = (𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩)
26	25	mptpreima 5545	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) “ 𝑥) = {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥}
27	17	toptopon 20548	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
28	15, 27	sylib 207	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑆 ∈ (TopOn‘∪ 𝑆))
29	16	toptopon 20548	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
30	13, 29	sylib 207	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑅 ∈ (TopOn‘∪ 𝑅))
31		xp1st 7089	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ (∪ 𝑅 × ∪ 𝑆) → (1st ‘𝑤) ∈ ∪ 𝑅)
32	20, 31	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (1st ‘𝑤) ∈ ∪ 𝑅)
33	28, 30, 32	cnmptc 21275	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑆 ↦ (1st ‘𝑤)) ∈ (𝑆 Cn 𝑅))
34	28	cnmptid 21274	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑆 ↦ 𝑎) ∈ (𝑆 Cn 𝑆))
35	28, 33, 34	cnmpt1t 21278	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) ∈ (𝑆 Cn (𝑅 ×t 𝑆)))
36		simplr 788	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))))
37	6, 36	sseldi 3566	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑥 ∈ (𝑅 ×t 𝑆))
38		cnima 20879	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) ∈ (𝑆 Cn (𝑅 ×t 𝑆)) ∧ 𝑥 ∈ (𝑅 ×t 𝑆)) → (◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) “ 𝑥) ∈ 𝑆)
39	35, 37, 38	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) “ 𝑥) ∈ 𝑆)
40	26, 39	syl5eqelr 2693	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ∈ 𝑆)
41		simprl 790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑧 ∈ 𝑥)
42		elunii 4377	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑥 ∈ (𝑅 ×t 𝑆)) → 𝑧 ∈ ∪ (𝑅 ×t 𝑆))
43	41, 37, 42	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑧 ∈ ∪ (𝑅 ×t 𝑆))
44	43, 19	eleqtrrd 2691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑧 ∈ (∪ 𝑅 × ∪ 𝑆))
45		xp2nd 7090	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (∪ 𝑅 × ∪ 𝑆) → (2nd ‘𝑧) ∈ ∪ 𝑆)
46	44, 45	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (2nd ‘𝑧) ∈ ∪ 𝑆)
47		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) = (𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩)
48	47	mptpreima 5545	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) “ 𝑥) = {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥}
49	30	cnmptid 21274	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑅 ↦ 𝑎) ∈ (𝑅 Cn 𝑅))
50	30, 28, 46	cnmptc 21275	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑅 ↦ (2nd ‘𝑧)) ∈ (𝑅 Cn 𝑆))
51	30, 49, 50	cnmpt1t 21278	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) ∈ (𝑅 Cn (𝑅 ×t 𝑆)))
52		cnima 20879	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) ∈ (𝑅 Cn (𝑅 ×t 𝑆)) ∧ 𝑥 ∈ (𝑅 ×t 𝑆)) → (◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) “ 𝑥) ∈ 𝑅)
53	51, 37, 52	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) “ 𝑥) ∈ 𝑅)
54	48, 53	syl5eqelr 2693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} ∈ 𝑅)
55		xp1st 7089	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ (∪ 𝑅 × ∪ 𝑆) → (1st ‘𝑧) ∈ ∪ 𝑅)
56	44, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (1st ‘𝑧) ∈ ∪ 𝑅)
57		1st2nd2 7096	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ (∪ 𝑅 × ∪ 𝑆) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
58	44, 57	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
59	58, 41	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ 𝑥)
60		opeq1 4340	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (1st ‘𝑧) → ⟨𝑎, (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
61	60	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (1st ‘𝑧) → (⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥 ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ 𝑥))
62	61	rspcev 3282	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1st ‘𝑧) ∈ ∪ 𝑅 ∧ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ 𝑥) → ∃𝑎 ∈ ∪ 𝑅⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥)
63	56, 59, 62	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → ∃𝑎 ∈ ∪ 𝑅⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥)
64		rabn0 3912	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} ≠ ∅ ↔ ∃𝑎 ∈ ∪ 𝑅⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥)
65	63, 64	sylibr 223	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} ≠ ∅)
66		inss2 3796	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) ⊆ (Clsd‘(𝑅 ×t 𝑆))
67	66, 36	sseldi 3566	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑥 ∈ (Clsd‘(𝑅 ×t 𝑆)))
68		cnclima 20882	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) ∈ (𝑅 Cn (𝑅 ×t 𝑆)) ∧ 𝑥 ∈ (Clsd‘(𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) “ 𝑥) ∈ (Clsd‘𝑅))
69	51, 67, 68	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑅 ↦ ⟨𝑎, (2nd ‘𝑧)⟩) “ 𝑥) ∈ (Clsd‘𝑅))
70	48, 69	syl5eqelr 2693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} ∈ (Clsd‘𝑅))
71	16, 12, 54, 65, 70	conclo 21028	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} = ∪ 𝑅)
72	32, 71	eleqtrrd 2691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (1st ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥})
73		opeq1 4340	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (1st ‘𝑤) → ⟨𝑎, (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩)
74	73	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (1st ‘𝑤) → (⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥))
75	74	elrab 3331	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} ↔ ((1st ‘𝑤) ∈ ∪ 𝑅 ∧ ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥))
76	75	simprbi 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅 ∣ ⟨𝑎, (2nd ‘𝑧)⟩ ∈ 𝑥} → ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥)
77	72, 76	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥)
78		opeq2 4341	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (2nd ‘𝑧) → ⟨(1st ‘𝑤), 𝑎⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩)
79	78	eleq1d 2672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (2nd ‘𝑧) → (⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥))
80	79	rspcev 3282	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘𝑧) ∈ ∪ 𝑆 ∧ ⟨(1st ‘𝑤), (2nd ‘𝑧)⟩ ∈ 𝑥) → ∃𝑎 ∈ ∪ 𝑆⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥)
81	46, 77, 80	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → ∃𝑎 ∈ ∪ 𝑆⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥)
82		rabn0 3912	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ≠ ∅ ↔ ∃𝑎 ∈ ∪ 𝑆⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥)
83	81, 82	sylibr 223	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ≠ ∅)
84		cnclima 20882	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) ∈ (𝑆 Cn (𝑅 ×t 𝑆)) ∧ 𝑥 ∈ (Clsd‘(𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) “ 𝑥) ∈ (Clsd‘𝑆))
85	35, 67, 84	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑆 ↦ ⟨(1st ‘𝑤), 𝑎⟩) “ 𝑥) ∈ (Clsd‘𝑆))
86	26, 85	syl5eqelr 2693	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ∈ (Clsd‘𝑆))
87	17, 14, 40, 83, 86	conclo 21028	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} = ∪ 𝑆)
88	24, 87	eleqtrrd 2691	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (2nd ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥})
89		opeq2 4341	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (2nd ‘𝑤) → ⟨(1st ‘𝑤), 𝑎⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
90	89	eleq1d 2672	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (2nd ‘𝑤) → (⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑥))
91	90	elrab 3331	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} ↔ ((2nd ‘𝑤) ∈ ∪ 𝑆 ∧ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑥))
92	91	simprbi 479	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆 ∣ ⟨(1st ‘𝑤), 𝑎⟩ ∈ 𝑥} → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑥)
93	88, 92	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑥)
94	22, 93	eqeltrd 2688	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑤 ∈ 𝑥)
95	94	expr 641	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → (𝑤 ∈ ∪ (𝑅 ×t 𝑆) → 𝑤 ∈ 𝑥))
96	95	ssrdv 3574	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → ∪ (𝑅 ×t 𝑆) ⊆ 𝑥)
97	10, 96	eqssd 3585	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 = ∪ (𝑅 ×t 𝑆))
98	97	ex 449	. . . . . . . 8 ⊢ (((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (𝑧 ∈ 𝑥 → 𝑥 = ∪ (𝑅 ×t 𝑆)))
99	98	exlimdv 1848	. . . . . . 7 ⊢ (((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (∃𝑧 𝑧 ∈ 𝑥 → 𝑥 = ∪ (𝑅 ×t 𝑆)))
100	5, 99	syl5bi 231	. . . . . 6 ⊢ (((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (¬ 𝑥 = ∅ → 𝑥 = ∪ (𝑅 ×t 𝑆)))
101	100	orrd 392	. . . . 5 ⊢ (((𝑅 ∈ Con ∧ 𝑆 ∈ Con) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×t 𝑆)))
102	101	ex 449	. . . 4 ⊢ ((𝑅 ∈ Con ∧ 𝑆 ∈ Con) → (𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) → (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×t 𝑆))))
103		vex 3176	. . . . 5 ⊢ 𝑥 ∈ V
104	103	elpr 4146	. . . 4 ⊢ (𝑥 ∈ {∅, ∪ (𝑅 ×t 𝑆)} ↔ (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×t 𝑆)))
105	102, 104	syl6ibr 241	. . 3 ⊢ ((𝑅 ∈ Con ∧ 𝑆 ∈ Con) → (𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) → 𝑥 ∈ {∅, ∪ (𝑅 ×t 𝑆)}))
106	105	ssrdv 3574	. 2 ⊢ ((𝑅 ∈ Con ∧ 𝑆 ∈ Con) → ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) ⊆ {∅, ∪ (𝑅 ×t 𝑆)})
107		eqid 2610	. . 3 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
108	107	iscon2 21027	. 2 ⊢ ((𝑅 ×t 𝑆) ∈ Con ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) ⊆ {∅, ∪ (𝑅 ×t 𝑆)}))
109	4, 106, 108	sylanbrc 695	1 ⊢ ((𝑅 ∈ Con ∧ 𝑆 ∈ Con) → (𝑅 ×t 𝑆) ∈ Con)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {cpr 4127  ⟨cop 4131  ∪ cuni 4372   ↦ cmpt 4643   × cxp 5036  ^◡ccnv 5037   “ cima 5041  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  Topctop 20517  TopOnctopon 20518  Clsdccld 20630   Cn ccn 20838  Conccon 21024   ×_t ctx 21173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-cn 20841  df-cnp 20842  df-con 21025  df-tx 21175

This theorem is referenced by:  cvmlift2lem9  30547  cvmlift2lem13  30551