txpconn - Mathbox for Mario Carneiro

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Theorem txpconn 34160

Description: The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)

**Assertion**
Ref	Expression
txpconn	⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×_t 𝑆) ∈ PConn)

Proof of Theorem txpconn Dummy variables 𝑓 𝑥 𝑦 𝑔 ℎ 𝑡 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pconntop 34153	. . 3 ⊢ (𝑅 ∈ PConn → 𝑅 ∈ Top)
2		pconntop 34153	. . 3 ⊢ (𝑆 ∈ PConn → 𝑆 ∈ Top)
3		txtop 23054	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×_t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 597	. 2 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×_t 𝑆) ∈ Top)
5		an6 1446	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PConn ∧ 𝑥 ∈ ∪ 𝑅 ∧ 𝑧 ∈ ∪ 𝑅) ∧ (𝑆 ∈ PConn ∧ 𝑦 ∈ ∪ 𝑆 ∧ 𝑤 ∈ ∪ 𝑆)) ↔ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)))
6		eqid 2733	. . . . . . . . . . . 12 ⊢ ∪ 𝑅 = ∪ 𝑅
7	6	pconncn 34152	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PConn ∧ 𝑥 ∈ ∪ 𝑅 ∧ 𝑧 ∈ ∪ 𝑅) → ∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
8		eqid 2733	. . . . . . . . . . . 12 ⊢ ∪ 𝑆 = ∪ 𝑆
9	8	pconncn 34152	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ PConn ∧ 𝑦 ∈ ∪ 𝑆 ∧ 𝑤 ∈ ∪ 𝑆) → ∃ℎ ∈ (II Cn 𝑆)((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤))
10	7, 9	anim12i 614	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PConn ∧ 𝑥 ∈ ∪ 𝑅 ∧ 𝑧 ∈ ∪ 𝑅) ∧ (𝑆 ∈ PConn ∧ 𝑦 ∈ ∪ 𝑆 ∧ 𝑤 ∈ ∪ 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ℎ ∈ (II Cn 𝑆)((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))
11	5, 10	sylbir 234	. . . . . . . . 9 ⊢ (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ℎ ∈ (II Cn 𝑆)((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))
12		reeanv 3227	. . . . . . . . 9 ⊢ (∃𝑔 ∈ (II Cn 𝑅)∃ℎ ∈ (II Cn 𝑆)(((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)) ↔ (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ℎ ∈ (II Cn 𝑆)((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))
13	11, 12	sylibr 233	. . . . . . . 8 ⊢ (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) → ∃𝑔 ∈ (II Cn 𝑅)∃ℎ ∈ (II Cn 𝑆)(((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))
14		iiuni 24378	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
15		eqid 2733	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)
16	14, 15	txcnmpt 23109	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) ∈ (II Cn (𝑅 ×_t 𝑆)))
17	16	ad2antrl 727	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) ∈ (II Cn (𝑅 ×_t 𝑆)))
18		0elunit 13441	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
19		fveq2 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 0 → (𝑔‘𝑡) = (𝑔‘0))
20		fveq2 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 0 → (ℎ‘𝑡) = (ℎ‘0))
21	19, 20	opeq12d 4879	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 0 → ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩ = ⟨(𝑔‘0), (ℎ‘0)⟩)
22		opex 5462	. . . . . . . . . . . . . 14 ⊢ ⟨(𝑔‘0), (ℎ‘0)⟩ ∈ V
23	21, 15, 22	fvmpt 6993	. . . . . . . . . . . . 13 ⊢ (0 ∈ (0[,]1) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0) = ⟨(𝑔‘0), (ℎ‘0)⟩)
24	18, 23	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0) = ⟨(𝑔‘0), (ℎ‘0)⟩
25		simprrl 780	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
26	25	simpld 496	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → (𝑔‘0) = 𝑥)
27		simprrr 781	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤))
28	27	simpld 496	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → (ℎ‘0) = 𝑦)
29	26, 28	opeq12d 4879	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ⟨(𝑔‘0), (ℎ‘0)⟩ = ⟨𝑥, 𝑦⟩)
30	24, 29	eqtrid 2785	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩)
31		1elunit 13442	. . . . . . . . . . . . 13 ⊢ 1 ∈ (0[,]1)
32		fveq2 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 1 → (𝑔‘𝑡) = (𝑔‘1))
33		fveq2 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 1 → (ℎ‘𝑡) = (ℎ‘1))
34	32, 33	opeq12d 4879	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 1 → ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩ = ⟨(𝑔‘1), (ℎ‘1)⟩)
35		opex 5462	. . . . . . . . . . . . . 14 ⊢ ⟨(𝑔‘1), (ℎ‘1)⟩ ∈ V
36	34, 15, 35	fvmpt 6993	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0[,]1) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1) = ⟨(𝑔‘1), (ℎ‘1)⟩)
37	31, 36	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1) = ⟨(𝑔‘1), (ℎ‘1)⟩
38	25	simprd 497	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → (𝑔‘1) = 𝑧)
39	27	simprd 497	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → (ℎ‘1) = 𝑤)
40	38, 39	opeq12d 4879	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ⟨(𝑔‘1), (ℎ‘1)⟩ = ⟨𝑧, 𝑤⟩)
41	37, 40	eqtrid 2785	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩)
42		fveq1 6886	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0))
43	42	eqeq1d 2735	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩))
44		fveq1 6886	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1))
45	44	eqeq1d 2735	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → ((𝑓‘1) = ⟨𝑧, 𝑤⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩))
46	43, 45	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → (((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ (((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩ ∧ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩)))
47	46	rspcev 3611	. . . . . . . . . . 11 ⊢ (((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) ∈ (II Cn (𝑅 ×_t 𝑆)) ∧ (((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩ ∧ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩)) → ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
48	17, 30, 41, 47	syl12anc 836	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
49	48	expr 458	. . . . . . . . 9 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ (𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆))) → ((((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)) → ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
50	49	rexlimdvva 3212	. . . . . . . 8 ⊢ (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)∃ℎ ∈ (II Cn 𝑆)(((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)) → ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
51	13, 50	mpd 15	. . . . . . 7 ⊢ (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) → ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
52	51	3expa 1119	. . . . . 6 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆)) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) → ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
53	52	ralrimivva 3201	. . . . 5 ⊢ (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆)) → ∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
54	53	ralrimivva 3201	. . . 4 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ∀𝑥 ∈ ∪ 𝑅∀𝑦 ∈ ∪ 𝑆∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
55		eqeq2 2745	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑓‘1) = 𝑣 ↔ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
56	55	anbi2d 630	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
57	56	rexbidv 3179	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
58	57	ralxp 5838	. . . . . 6 ⊢ (∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
59		eqeq2 2745	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑓‘0) = 𝑢 ↔ (𝑓‘0) = ⟨𝑥, 𝑦⟩))
60	59	anbi1d 631	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
61	60	rexbidv 3179	. . . . . . 7 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
62	61	2ralbidv 3219	. . . . . 6 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
63	58, 62	bitrid 283	. . . . 5 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
64	63	ralxp 5838	. . . 4 ⊢ (∀𝑢 ∈ (∪ 𝑅 × ∪ 𝑆)∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑥 ∈ ∪ 𝑅∀𝑦 ∈ ∪ 𝑆∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
65	54, 64	sylibr 233	. . 3 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ∀𝑢 ∈ (∪ 𝑅 × ∪ 𝑆)∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣))
66	6, 8	txuni 23077	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
67	1, 2, 66	syl2an 597	. . . 4 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
68	67	raleqdv 3326	. . . 4 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑣 ∈ ∪ (𝑅 ×_t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣)))
69	67, 68	raleqbidv 3343	. . 3 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (∀𝑢 ∈ (∪ 𝑅 × ∪ 𝑆)∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑢 ∈ ∪ (𝑅 ×_t 𝑆)∀𝑣 ∈ ∪ (𝑅 ×_t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣)))
70	65, 69	mpbid 231	. 2 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ∀𝑢 ∈ ∪ (𝑅 ×_t 𝑆)∀𝑣 ∈ ∪ (𝑅 ×_t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣))
71		eqid 2733	. . 3 ⊢ ∪ (𝑅 ×_t 𝑆) = ∪ (𝑅 ×_t 𝑆)
72	71	ispconn 34151	. 2 ⊢ ((𝑅 ×_t 𝑆) ∈ PConn ↔ ((𝑅 ×_t 𝑆) ∈ Top ∧ ∀𝑢 ∈ ∪ (𝑅 ×_t 𝑆)∀𝑣 ∈ ∪ (𝑅 ×_t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣)))
73	4, 70, 72	sylanbrc 584	1 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×_t 𝑆) ∈ PConn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ⟨cop 4632 ∪ cuni 4906 ↦ cmpt 5229 × cxp 5672 ‘cfv 6539 (class class class)co 7403 0cc0 11105 1c1 11106 [,]cicc 13322 Topctop 22376 Cn ccn 22709 ×_t ctx 23045 IIcii 24372 PConncpconn 34147

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-inf 9433 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-2 12270 df-3 12271 df-n0 12468 df-z 12554 df-uz 12818 df-q 12928 df-rp 12970 df-xneg 13087 df-xadd 13088 df-xmul 13089 df-icc 13326 df-seq 13962 df-exp 14023 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-topgen 17384 df-psmet 20920 df-xmet 20921 df-met 20922 df-bl 20923 df-mopn 20924 df-top 22377 df-topon 22394 df-bases 22430 df-cn 22712 df-tx 23047 df-ii 24374 df-pconn 34149

This theorem is referenced by: txsconn 34169

W3C validator