txpconn - Mathbox for Mario Carneiro

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Theorem txpconn 35732

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 35725	. . 3 ⊢ (𝑅 ∈ PConn → 𝑅 ∈ Top)
2		pconntop 35725	. . 3 ⊢ (𝑆 ∈ PConn → 𝑆 ∈ Top)
3		txtop 23737	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×_t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 607	. 2 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×_t 𝑆) ∈ Top)
5		an6 1473	. . . . . . . . . 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 2762	. . . . . . . . . . . 12 ⊢ ∪ 𝑅 = ∪ 𝑅
7	6	pconncn 35724	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PConn ∧ 𝑥 ∈ ∪ 𝑅 ∧ 𝑧 ∈ ∪ 𝑅) → ∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
8		eqid 2762	. . . . . . . . . . . 12 ⊢ ∪ 𝑆 = ∪ 𝑆
9	8	pconncn 35724	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ PConn ∧ 𝑦 ∈ ∪ 𝑆 ∧ 𝑤 ∈ ∪ 𝑆) → ∃ℎ ∈ (II Cn 𝑆)((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤))
10	7, 9	anim12i 624	. . . . . . . . . 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 238	. . . . . . . . 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 3236	. . . . . . . . 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 237	. . . . . . . 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 25051	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
15		eqid 2762	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)
16	14, 15	txcnmpt 23792	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) ∈ (II Cn (𝑅 ×_t 𝑆)))
17	16	ad2antrl 740	. . . . . . . . . . 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 13502	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
19		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 0 → (𝑔‘𝑡) = (𝑔‘0))
20		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 0 → (ℎ‘𝑡) = (ℎ‘0))
21	19, 20	opeq12d 4845	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 0 → ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩ = ⟨(𝑔‘0), (ℎ‘0)⟩)
22		opex 5444	. . . . . . . . . . . . . 14 ⊢ ⟨(𝑔‘0), (ℎ‘0)⟩ ∈ V
23	21, 15, 22	fvmpt 6989	. . . . . . . . . . . . 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 792	. . . . . . . . . . . . . 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 499	. . . . . . . . . . . . 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 793	. . . . . . . . . . . . . 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 499	. . . . . . . . . . . . 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 4845	. . . . . . . . . . . 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 2809	. . . . . . . . . . 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 13503	. . . . . . . . . . . . 13 ⊢ 1 ∈ (0[,]1)
32		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 1 → (𝑔‘𝑡) = (𝑔‘1))
33		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 1 → (ℎ‘𝑡) = (ℎ‘1))
34	32, 33	opeq12d 4845	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 1 → ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩ = ⟨(𝑔‘1), (ℎ‘1)⟩)
35		opex 5444	. . . . . . . . . . . . . 14 ⊢ ⟨(𝑔‘1), (ℎ‘1)⟩ ∈ V
36	34, 15, 35	fvmpt 6989	. . . . . . . . . . . . 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 500	. . . . . . . . . . . . 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 500	. . . . . . . . . . . . 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 4845	. . . . . . . . . . . 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 2809	. . . . . . . . . . 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 6880	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0))
43	42	eqeq1d 2764	. . . . . . . . . . . . 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 6880	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1))
45	44	eqeq1d 2764	. . . . . . . . . . . . 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 643	. . . . . . . . . . . 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 3580	. . . . . . . . . . 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 849	. . . . . . . . . 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 461	. . . . . . . . 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 3221	. . . . . . . 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 16	. . . . . . 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 1135	. . . . . 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 3207	. . . . 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 3207	. . . 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 2774	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑓‘1) = 𝑣 ↔ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
56	55	anbi2d 641	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
57	56	rexbidv 3188	. . . . . . 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 5826	. . . . . 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 2774	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑓‘0) = 𝑢 ↔ (𝑓‘0) = ⟨𝑥, 𝑦⟩))
60	59	anbi1d 642	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
61	60	rexbidv 3188	. . . . . . 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 3228	. . . . . 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 286	. . . . 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 5826	. . . 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 237	. . 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 23760	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
67	1, 2, 66	syl2an 607	. . . 4 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
68	67	raleqdv 3322	. . . 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 3337	. . 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 235	. 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 2762	. . 3 ⊢ ∪ (𝑅 ×_t 𝑆) = ∪ (𝑅 ×_t 𝑆)
72	71	ispconn 35723	. 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 594	1 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×_t 𝑆) ∈ PConn)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ∀wral 3078 ∃wrex 3088 ⟨cop 4594 ∪ cuni 4871 ↦ cmpt 5191 × cxp 5658 ‘cfv 6536 (class class class)co 7412 0cc0 11106 1c1 11107 [,]cicc 13381 Topctop 23061 Cn ccn 23392 ×_t ctx 23728 IIcii 25045 PConncpconn 35719

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-er 8692 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9400 df-inf 9401 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-n0 12511 df-z 12598 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-icc 13385 df-seq 14045 df-exp 14105 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-topgen 17502 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-top 23062 df-topon 23079 df-bases 23114 df-cn 23395 df-tx 23730 df-ii 25047 df-pconn 35721

This theorem is used by: txsconn 35741

W3C validator