txpconn - Mathbox for Mario Carneiro

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Theorem txpconn 33826

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 33819	. . 3 ⊢ (𝑅 ∈ PConn → 𝑅 ∈ Top)
2		pconntop 33819	. . 3 ⊢ (𝑆 ∈ PConn → 𝑆 ∈ Top)
3		txtop 22920	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×_t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×_t 𝑆) ∈ Top)
5		an6 1445	. . . . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ ∪ 𝑅 = ∪ 𝑅
7	6	pconncn 33818	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PConn ∧ 𝑥 ∈ ∪ 𝑅 ∧ 𝑧 ∈ ∪ 𝑅) → ∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
8		eqid 2736	. . . . . . . . . . . 12 ⊢ ∪ 𝑆 = ∪ 𝑆
9	8	pconncn 33818	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ PConn ∧ 𝑦 ∈ ∪ 𝑆 ∧ 𝑤 ∈ ∪ 𝑆) → ∃ℎ ∈ (II Cn 𝑆)((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤))
10	7, 9	anim12i 613	. . . . . . . . . 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 3217	. . . . . . . . 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 24244	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
15		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)
16	14, 15	txcnmpt 22975	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) ∈ (II Cn (𝑅 ×_t 𝑆)))
17	16	ad2antrl 726	. . . . . . . . . . 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 13386	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
19		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 0 → (𝑔‘𝑡) = (𝑔‘0))
20		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 0 → (ℎ‘𝑡) = (ℎ‘0))
21	19, 20	opeq12d 4838	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 0 → ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩ = ⟨(𝑔‘0), (ℎ‘0)⟩)
22		opex 5421	. . . . . . . . . . . . . 14 ⊢ ⟨(𝑔‘0), (ℎ‘0)⟩ ∈ V
23	21, 15, 22	fvmpt 6948	. . . . . . . . . . . . 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 779	. . . . . . . . . . . . . 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 495	. . . . . . . . . . . . 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 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)))) → ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤))
28	27	simpld 495	. . . . . . . . . . . . 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 4838	. . . . . . . . . . . 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 2788	. . . . . . . . . . 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 13387	. . . . . . . . . . . . 13 ⊢ 1 ∈ (0[,]1)
32		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 1 → (𝑔‘𝑡) = (𝑔‘1))
33		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 1 → (ℎ‘𝑡) = (ℎ‘1))
34	32, 33	opeq12d 4838	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 1 → ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩ = ⟨(𝑔‘1), (ℎ‘1)⟩)
35		opex 5421	. . . . . . . . . . . . . 14 ⊢ ⟨(𝑔‘1), (ℎ‘1)⟩ ∈ V
36	34, 15, 35	fvmpt 6948	. . . . . . . . . . . . 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 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‘1) = 𝑧)
39	27	simprd 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‘1) = 𝑤)
40	38, 39	opeq12d 4838	. . . . . . . . . . . 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 2788	. . . . . . . . . . 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 6841	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0))
43	42	eqeq1d 2738	. . . . . . . . . . . . 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 6841	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1))
45	44	eqeq1d 2738	. . . . . . . . . . . . 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 631	. . . . . . . . . . . 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 3581	. . . . . . . . . . 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 835	. . . . . . . . . 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 457	. . . . . . . . 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 3205	. . . . . . . 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 1118	. . . . . 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 3197	. . . . 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 3197	. . . 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 2748	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑓‘1) = 𝑣 ↔ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
56	55	anbi2d 629	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
57	56	rexbidv 3175	. . . . . . 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 5797	. . . . . 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 2748	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑓‘0) = 𝑢 ↔ (𝑓‘0) = ⟨𝑥, 𝑦⟩))
60	59	anbi1d 630	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
61	60	rexbidv 3175	. . . . . . 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 3212	. . . . . 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 282	. . . . 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 5797	. . . 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 22943	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
67	1, 2, 66	syl2an 596	. . . 4 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
68	67	raleqdv 3313	. . . 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 3319	. . 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 2736	. . 3 ⊢ ∪ (𝑅 ×_t 𝑆) = ∪ (𝑅 ×_t 𝑆)
72	71	ispconn 33817	. 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 583	1 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×_t 𝑆) ∈ PConn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 ⟨cop 4592 ∪ cuni 4865 ↦ cmpt 5188 × cxp 5631 ‘cfv 6496 (class class class)co 7357 0cc0 11051 1c1 11052 [,]cicc 13267 Topctop 22242 Cn ccn 22575 ×_t ctx 22911 IIcii 24238 PConncpconn 33813

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-icc 13271 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-topgen 17325 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-top 22243 df-topon 22260 df-bases 22296 df-cn 22578 df-tx 22913 df-ii 24240 df-pconn 33815

This theorem is referenced by: txsconn 33835

W3C validator