txpcon - Mathbox for Mario Carneiro

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Theorem txpcon 30468

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

**Assertion**
Ref	Expression
txpcon	⊢ ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → (𝑅 ×_t 𝑆) ∈ PCon)

Proof of Theorem txpcon 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		pcontop 30461	. . 3 ⊢ (𝑅 ∈ PCon → 𝑅 ∈ Top)
2		pcontop 30461	. . 3 ⊢ (𝑆 ∈ PCon → 𝑆 ∈ Top)
3		txtop 21182	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×_t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 493	. 2 ⊢ ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → (𝑅 ×_t 𝑆) ∈ Top)
5		an6 1400	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PCon ∧ 𝑥 ∈ ∪ 𝑅 ∧ 𝑧 ∈ ∪ 𝑅) ∧ (𝑆 ∈ PCon ∧ 𝑦 ∈ ∪ 𝑆 ∧ 𝑤 ∈ ∪ 𝑆)) ↔ ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)))
6		eqid 2610	. . . . . . . . . . . 12 ⊢ ∪ 𝑅 = ∪ 𝑅
7	6	pconcn 30460	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PCon ∧ 𝑥 ∈ ∪ 𝑅 ∧ 𝑧 ∈ ∪ 𝑅) → ∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
8		eqid 2610	. . . . . . . . . . . 12 ⊢ ∪ 𝑆 = ∪ 𝑆
9	8	pconcn 30460	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ PCon ∧ 𝑦 ∈ ∪ 𝑆 ∧ 𝑤 ∈ ∪ 𝑆) → ∃ℎ ∈ (II Cn 𝑆)((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤))
10	7, 9	anim12i 588	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PCon ∧ 𝑥 ∈ ∪ 𝑅 ∧ 𝑧 ∈ ∪ 𝑅) ∧ (𝑆 ∈ PCon ∧ 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 224	. . . . . . . . 9 ⊢ (((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 3086	. . . . . . . . 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 223	. . . . . . . 8 ⊢ (((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 22492	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
15		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)
16	14, 15	txcnmpt 21237	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) ∈ (II Cn (𝑅 ×_t 𝑆)))
17	16	ad2antrl 760	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 12161	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
19		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 0 → (𝑔‘𝑡) = (𝑔‘0))
20		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 0 → (ℎ‘𝑡) = (ℎ‘0))
21	19, 20	opeq12d 4348	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 0 → ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩ = ⟨(𝑔‘0), (ℎ‘0)⟩)
22		opex 4859	. . . . . . . . . . . . . 14 ⊢ ⟨(𝑔‘0), (ℎ‘0)⟩ ∈ V
23	21, 15, 22	fvmpt 6191	. . . . . . . . . . . . 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 800	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 474	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 801	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 474	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 4348	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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	syl5eq 2656	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 12162	. . . . . . . . . . . . 13 ⊢ 1 ∈ (0[,]1)
32		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 1 → (𝑔‘𝑡) = (𝑔‘1))
33		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 1 → (ℎ‘𝑡) = (ℎ‘1))
34	32, 33	opeq12d 4348	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 1 → ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩ = ⟨(𝑔‘1), (ℎ‘1)⟩)
35		opex 4859	. . . . . . . . . . . . . 14 ⊢ ⟨(𝑔‘1), (ℎ‘1)⟩ ∈ V
36	34, 15, 35	fvmpt 6191	. . . . . . . . . . . . 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 478	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 478	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 4348	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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	syl5eq 2656	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 6102	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0))
43	42	eqeq1d 2612	. . . . . . . . . . . . 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 6102	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1))
45	44	eqeq1d 2612	. . . . . . . . . . . . 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 743	. . . . . . . . . . . 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 3282	. . . . . . . . . . 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 1316	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 641	. . . . . . . . 9 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 3020	. . . . . . . 8 ⊢ (((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 1257	. . . . . 6 ⊢ ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 2954	. . . . 5 ⊢ (((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 2954	. . . 4 ⊢ ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → ∀𝑥 ∈ ∪ 𝑅∀𝑦 ∈ ∪ 𝑆∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
55		eqeq2 2621	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑓‘1) = 𝑣 ↔ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
56	55	anbi2d 736	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
57	56	rexbidv 3034	. . . . . . 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 5185	. . . . . 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 2621	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑓‘0) = 𝑢 ↔ (𝑓‘0) = ⟨𝑥, 𝑦⟩))
60	59	anbi1d 737	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
61	60	rexbidv 3034	. . . . . . 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 2972	. . . . . 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	syl5bb 271	. . . . 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 5185	. . . 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 223	. . 3 ⊢ ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → ∀𝑢 ∈ (∪ 𝑅 × ∪ 𝑆)∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣))
66	6, 8	txuni 21205	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
67	1, 2, 66	syl2an 493	. . . 4 ⊢ ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
68	67	raleqdv 3121	. . . 4 ⊢ ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → (∀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 3129	. . 3 ⊢ ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → (∀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 221	. 2 ⊢ ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → ∀𝑢 ∈ ∪ (𝑅 ×_t 𝑆)∀𝑣 ∈ ∪ (𝑅 ×_t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣))
71		eqid 2610	. . 3 ⊢ ∪ (𝑅 ×_t 𝑆) = ∪ (𝑅 ×_t 𝑆)
72	71	ispcon 30459	. 2 ⊢ ((𝑅 ×_t 𝑆) ∈ PCon ↔ ((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 695	1 ⊢ ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → (𝑅 ×_t 𝑆) ∈ PCon)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⟨cop 4131 ∪ cuni 4372 ↦ cmpt 4643 × cxp 5036 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 [,]cicc 12049 Topctop 20517 Cn ccn 20838 ×_t ctx 21173 IIcii 22486 PConcpcon 30455

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 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cn 20841 df-tx 21175 df-ii 22488 df-pcon 30457

This theorem is referenced by: txscon 30477

W3C validator