Theorem txtube 22791

Description: The "tube lemma". If 𝑋 is compact and there is an open set 𝑈 containing the line 𝑋 × {𝐴}, then there is a "tube" 𝑋 × 𝑢 for some neighborhood 𝑢 of 𝐴 which is entirely contained within 𝑈. (Contributed by Mario Carneiro, 21-Mar-2015.)

Hypotheses

Ref	Expression
txtube.x	⊢ 𝑋 = ∪ 𝑅
txtube.y	⊢ 𝑌 = ∪ 𝑆
txtube.r	⊢ (𝜑 → 𝑅 ∈ Comp)
txtube.s	⊢ (𝜑 → 𝑆 ∈ Top)
txtube.w	⊢ (𝜑 → 𝑈 ∈ (𝑅 ×t 𝑆))
txtube.u	⊢ (𝜑 → (𝑋 × {𝐴}) ⊆ 𝑈)
txtube.a	⊢ (𝜑 → 𝐴 ∈ 𝑌)

Assertion

Ref	Expression
txtube	⊢ (𝜑 → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))

Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑆   𝑢,𝑌   𝜑,𝑢   𝑢,𝑈   𝑢,𝑋

Proof of Theorem txtube

Dummy variables 𝑡 𝑓 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		txtube.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Comp)
2		eleq1 2826	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑥, 𝐴⟩ → (𝑦 ∈ (𝑢 × 𝑣) ↔ ⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣)))
3	2	anbi1d 630	. . . . . . 7 ⊢ (𝑦 = ⟨𝑥, 𝐴⟩ → ((𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
4	3	2rexbidv 3229	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 𝐴⟩ → (∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
5		txtube.w	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (𝑅 ×t 𝑆))
6		txtube.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Top)
7		eltx 22719	. . . . . . . . 9 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Top) → (𝑈 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
8	1, 6, 7	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
9	5, 8	mpbid 231	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
10	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
11		txtube.u	. . . . . . . 8 ⊢ (𝜑 → (𝑋 × {𝐴}) ⊆ 𝑈)
12	11	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑋 × {𝐴}) ⊆ 𝑈)
13		id 22	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝑋)
14		txtube.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑌)
15		snidg 4595	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑌 → 𝐴 ∈ {𝐴})
16	14, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
17		opelxpi 5626	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ {𝐴}) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × {𝐴}))
18	13, 16, 17	syl2anr 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × {𝐴}))
19	12, 18	sseldd 3922	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝑥, 𝐴⟩ ∈ 𝑈)
20	4, 10, 19	rspcdva 3562	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
21		opelxp 5625	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ↔ (𝑥 ∈ 𝑢 ∧ 𝐴 ∈ 𝑣))
22	21	anbi1i 624	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ((𝑥 ∈ 𝑢 ∧ 𝐴 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
23		anass 469	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑢 ∧ 𝐴 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥 ∈ 𝑢 ∧ (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
24	22, 23	bitri 274	. . . . . . . 8 ⊢ ((⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥 ∈ 𝑢 ∧ (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
25	24	rexbii 3181	. . . . . . 7 ⊢ (∃𝑣 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑣 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
26		r19.42v 3279	. . . . . . 7 ⊢ (∃𝑣 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)) ↔ (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑆 (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
27	25, 26	bitri 274	. . . . . 6 ⊢ (∃𝑣 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑆 (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
28	27	rexbii 3181	. . . . 5 ⊢ (∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑢 ∈ 𝑅 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑆 (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
29	20, 28	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑢 ∈ 𝑅 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑆 (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
30	29	ralrimiva 3103	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑅 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑆 (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
31		txtube.x	. . . 4 ⊢ 𝑋 = ∪ 𝑅
32		eleq2 2827	. . . . 5 ⊢ (𝑣 = (𝑓‘𝑢) → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ (𝑓‘𝑢)))
33		xpeq2 5610	. . . . . 6 ⊢ (𝑣 = (𝑓‘𝑢) → (𝑢 × 𝑣) = (𝑢 × (𝑓‘𝑢)))
34	33	sseq1d 3952	. . . . 5 ⊢ (𝑣 = (𝑓‘𝑢) → ((𝑢 × 𝑣) ⊆ 𝑈 ↔ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈))
35	32, 34	anbi12d 631	. . . 4 ⊢ (𝑣 = (𝑓‘𝑢) → ((𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))
36	31, 35	cmpcovf 22542	. . 3 ⊢ ((𝑅 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑅 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑆 (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈))) → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = ∪ 𝑡 ∧ ∃𝑓(𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈))))
37	1, 30, 36	syl2anc 584	. 2 ⊢ (𝜑 → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = ∪ 𝑡 ∧ ∃𝑓(𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈))))
38		rint0 4921	. . . . . . . . . 10 ⊢ (ran 𝑓 = ∅ → (𝑌 ∩ ∩ ran 𝑓) = 𝑌)
39	38	adantl 482	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 = ∅) → (𝑌 ∩ ∩ ran 𝑓) = 𝑌)
40		txtube.y	. . . . . . . . . . . 12 ⊢ 𝑌 = ∪ 𝑆
41	40	topopn 22055	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Top → 𝑌 ∈ 𝑆)
42	6, 41	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
43	42	ad3antrrr 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 = ∅) → 𝑌 ∈ 𝑆)
44	39, 43	eqeltrd 2839	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 = ∅) → (𝑌 ∩ ∩ ran 𝑓) ∈ 𝑆)
45	6	ad3antrrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → 𝑆 ∈ Top)
46		simprrl 778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝑓:𝑡⟶𝑆)
47	46	frnd 6608	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → ran 𝑓 ⊆ 𝑆)
48	47	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓 ⊆ 𝑆)
49		simpr 485	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓 ≠ ∅)
50		simplr 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝑡 ∈ (𝒫 𝑅 ∩ Fin))
51	50	elin2d 4133	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝑡 ∈ Fin)
52	46	ffnd 6601	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝑓 Fn 𝑡)
53		dffn4 6694	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn 𝑡 ↔ 𝑓:𝑡–onto→ran 𝑓)
54	52, 53	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝑓:𝑡–onto→ran 𝑓)
55		fofi 9105	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ Fin ∧ 𝑓:𝑡–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
56	51, 54, 55	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → ran 𝑓 ∈ Fin)
57	56	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓 ∈ Fin)
58		fiinopn 22050	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ Top → ((ran 𝑓 ⊆ 𝑆 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin) → ∩ ran 𝑓 ∈ 𝑆))
59	58	imp 407	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Top ∧ (ran 𝑓 ⊆ 𝑆 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin)) → ∩ ran 𝑓 ∈ 𝑆)
60	45, 48, 49, 57, 59	syl13anc 1371	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ∩ ran 𝑓 ∈ 𝑆)
61		elssuni 4871	. . . . . . . . . . . 12 ⊢ (∩ ran 𝑓 ∈ 𝑆 → ∩ ran 𝑓 ⊆ ∪ 𝑆)
62	60, 61	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ∩ ran 𝑓 ⊆ ∪ 𝑆)
63	62, 40	sseqtrrdi 3972	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ∩ ran 𝑓 ⊆ 𝑌)
64		sseqin2 4149	. . . . . . . . . 10 ⊢ (∩ ran 𝑓 ⊆ 𝑌 ↔ (𝑌 ∩ ∩ ran 𝑓) = ∩ ran 𝑓)
65	63, 64	sylib 217	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → (𝑌 ∩ ∩ ran 𝑓) = ∩ ran 𝑓)
66	65, 60	eqeltrd 2839	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → (𝑌 ∩ ∩ ran 𝑓) ∈ 𝑆)
67	44, 66	pm2.61dane 3032	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → (𝑌 ∩ ∩ ran 𝑓) ∈ 𝑆)
68	14	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝐴 ∈ 𝑌)
69		simprrr 779	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈))
70		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈) → 𝐴 ∈ (𝑓‘𝑢))
71	70	ralimi 3087	. . . . . . . . . . 11 ⊢ (∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈) → ∀𝑢 ∈ 𝑡 𝐴 ∈ (𝑓‘𝑢))
72	69, 71	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → ∀𝑢 ∈ 𝑡 𝐴 ∈ (𝑓‘𝑢))
73		eliin 4929	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑌 → (𝐴 ∈ ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) ↔ ∀𝑢 ∈ 𝑡 𝐴 ∈ (𝑓‘𝑢)))
74	68, 73	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → (𝐴 ∈ ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) ↔ ∀𝑢 ∈ 𝑡 𝐴 ∈ (𝑓‘𝑢)))
75	72, 74	mpbird 256	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝐴 ∈ ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢))
76		fniinfv 6846	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑡 → ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) = ∩ ran 𝑓)
77	52, 76	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) = ∩ ran 𝑓)
78	75, 77	eleqtrd 2841	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝐴 ∈ ∩ ran 𝑓)
79	68, 78	elind 4128	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝐴 ∈ (𝑌 ∩ ∩ ran 𝑓))
80		simprl 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝑋 = ∪ 𝑡)
81		uniiun 4988	. . . . . . . . . . 11 ⊢ ∪ 𝑡 = ∪ 𝑢 ∈ 𝑡 𝑢
82	80, 81	eqtrdi 2794	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝑋 = ∪ 𝑢 ∈ 𝑡 𝑢)
83	82	xpeq1d 5618	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → (𝑋 × (𝑌 ∩ ∩ ran 𝑓)) = (∪ 𝑢 ∈ 𝑡 𝑢 × (𝑌 ∩ ∩ ran 𝑓)))
84		xpiundir 5658	. . . . . . . . 9 ⊢ (∪ 𝑢 ∈ 𝑡 𝑢 × (𝑌 ∩ ∩ ran 𝑓)) = ∪ 𝑢 ∈ 𝑡 (𝑢 × (𝑌 ∩ ∩ ran 𝑓))
85	83, 84	eqtrdi 2794	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → (𝑋 × (𝑌 ∩ ∩ ran 𝑓)) = ∪ 𝑢 ∈ 𝑡 (𝑢 × (𝑌 ∩ ∩ ran 𝑓)))
86		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈) → (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)
87	86	ralimi 3087	. . . . . . . . . . 11 ⊢ (∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈) → ∀𝑢 ∈ 𝑡 (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)
88	69, 87	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → ∀𝑢 ∈ 𝑡 (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)
89		inss2 4163	. . . . . . . . . . . . 13 ⊢ (𝑌 ∩ ∩ ran 𝑓) ⊆ ∩ ran 𝑓
90	76	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) = ∩ ran 𝑓)
91		iinss2 4987	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑡 → ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) ⊆ (𝑓‘𝑢))
92	91	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) ⊆ (𝑓‘𝑢))
93	90, 92	eqsstrrd 3960	. . . . . . . . . . . . 13 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → ∩ ran 𝑓 ⊆ (𝑓‘𝑢))
94	89, 93	sstrid 3932	. . . . . . . . . . . 12 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → (𝑌 ∩ ∩ ran 𝑓) ⊆ (𝑓‘𝑢))
95		xpss2 5609	. . . . . . . . . . . 12 ⊢ ((𝑌 ∩ ∩ ran 𝑓) ⊆ (𝑓‘𝑢) → (𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ (𝑢 × (𝑓‘𝑢)))
96		sstr2 3928	. . . . . . . . . . . 12 ⊢ ((𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ (𝑢 × (𝑓‘𝑢)) → ((𝑢 × (𝑓‘𝑢)) ⊆ 𝑈 → (𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈))
97	94, 95, 96	3syl 18	. . . . . . . . . . 11 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → ((𝑢 × (𝑓‘𝑢)) ⊆ 𝑈 → (𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈))
98	97	ralimdva 3108	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑡 → (∀𝑢 ∈ 𝑡 (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈 → ∀𝑢 ∈ 𝑡 (𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈))
99	52, 88, 98	sylc 65	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → ∀𝑢 ∈ 𝑡 (𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈)
100		iunss 4975	. . . . . . . . 9 ⊢ (∪ 𝑢 ∈ 𝑡 (𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈 ↔ ∀𝑢 ∈ 𝑡 (𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈)
101	99, 100	sylibr 233	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → ∪ 𝑢 ∈ 𝑡 (𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈)
102	85, 101	eqsstrd 3959	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → (𝑋 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈)
103		eleq2 2827	. . . . . . . . 9 ⊢ (𝑢 = (𝑌 ∩ ∩ ran 𝑓) → (𝐴 ∈ 𝑢 ↔ 𝐴 ∈ (𝑌 ∩ ∩ ran 𝑓)))
104		xpeq2 5610	. . . . . . . . . 10 ⊢ (𝑢 = (𝑌 ∩ ∩ ran 𝑓) → (𝑋 × 𝑢) = (𝑋 × (𝑌 ∩ ∩ ran 𝑓)))
105	104	sseq1d 3952	. . . . . . . . 9 ⊢ (𝑢 = (𝑌 ∩ ∩ ran 𝑓) → ((𝑋 × 𝑢) ⊆ 𝑈 ↔ (𝑋 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈))
106	103, 105	anbi12d 631	. . . . . . . 8 ⊢ (𝑢 = (𝑌 ∩ ∩ ran 𝑓) → ((𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈) ↔ (𝐴 ∈ (𝑌 ∩ ∩ ran 𝑓) ∧ (𝑋 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈)))
107	106	rspcev 3561	. . . . . . 7 ⊢ (((𝑌 ∩ ∩ ran 𝑓) ∈ 𝑆 ∧ (𝐴 ∈ (𝑌 ∩ ∩ ran 𝑓) ∧ (𝑋 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈)) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
108	67, 79, 102, 107	syl12anc 834	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
109	108	expr 457	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = ∪ 𝑡) → ((𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
110	109	exlimdv 1936	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = ∪ 𝑡) → (∃𝑓(𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
111	110	expimpd 454	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) → ((𝑋 = ∪ 𝑡 ∧ ∃𝑓(𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈))) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
112	111	rexlimdva 3213	. 2 ⊢ (𝜑 → (∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = ∪ 𝑡 ∧ ∃𝑓(𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈))) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
113	37, 112	mpd 15	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  ⟨cop 4567  ∪ cuni 4839  ∩ cint 4879  ∪ ciun 4924  ∩ ciin 4925   × cxp 5587  ran crn 5590   Fn wfn 6428  ⟶wf 6429  –onto→wfo 6431  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  Topctop 22042  Compccmp 22537   ×_t ctx 22711

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-fin 8737  df-topgen 17154  df-top 22043  df-cmp 22538  df-tx 22713

This theorem is referenced by:  txcmplem1  22792  xkoinjcn  22838  cvmlift2lem12  33276