Theorem txtube 23669

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 2832	. . . . . . . 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 3228	. . . . . 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 23597	. . . . . . . . 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 583	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
9	5, 8	mpbid 232	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
10	9	adantr 480	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑋 × {𝐴}) ⊆ 𝑈)
13		id 22	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝑋)
14		txtube.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑌)
15		snidg 4682	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑌 → 𝐴 ∈ {𝐴})
16	14, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
17		opelxpi 5737	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ {𝐴}) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × {𝐴}))
18	13, 16, 17	syl2anr 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × {𝐴}))
19	12, 18	sseldd 4009	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝑥, 𝐴⟩ ∈ 𝑈)
20	4, 10, 19	rspcdva 3636	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
21		opelxp 5736	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ↔ (𝑥 ∈ 𝑢 ∧ 𝐴 ∈ 𝑣))
22	21	anbi1i 623	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ((𝑥 ∈ 𝑢 ∧ 𝐴 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
23		anass 468	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑢 ∧ 𝐴 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥 ∈ 𝑢 ∧ (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
24	22, 23	bitri 275	. . . . . . . 8 ⊢ ((⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥 ∈ 𝑢 ∧ (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
25	24	rexbii 3100	. . . . . . 7 ⊢ (∃𝑣 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑣 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
26		r19.42v 3197	. . . . . . 7 ⊢ (∃𝑣 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)) ↔ (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑆 (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
27	25, 26	bitri 275	. . . . . 6 ⊢ (∃𝑣 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑆 (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
28	27	rexbii 3100	. . . . 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 218	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑢 ∈ 𝑅 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑆 (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
30	29	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑅 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑆 (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
31		txtube.x	. . . 4 ⊢ 𝑋 = ∪ 𝑅
32		eleq2 2833	. . . . 5 ⊢ (𝑣 = (𝑓‘𝑢) → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ (𝑓‘𝑢)))
33		xpeq2 5721	. . . . . 6 ⊢ (𝑣 = (𝑓‘𝑢) → (𝑢 × 𝑣) = (𝑢 × (𝑓‘𝑢)))
34	33	sseq1d 4040	. . . . 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 23420	. . 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 583	. 2 ⊢ (𝜑 → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = ∪ 𝑡 ∧ ∃𝑓(𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈))))
38		rint0 5012	. . . . . . . . . 10 ⊢ (ran 𝑓 = ∅ → (𝑌 ∩ ∩ ran 𝑓) = 𝑌)
39	38	adantl 481	. . . . . . . . 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 22933	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Top → 𝑌 ∈ 𝑆)
42	6, 41	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
43	42	ad3antrrr 729	. . . . . . . . 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 2844	. . . . . . . 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 729	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 6755	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 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 4228	. . . . . . . . . . . . . . 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 6748	. . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn 𝑡 ↔ 𝑓:𝑡–onto→ran 𝑓)
54	52, 53	sylib 218	. . . . . . . . . . . . . . 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 9379	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ Fin ∧ 𝑓:𝑡–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
56	51, 54, 55	syl2anc 583	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 22928	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ Top → ((ran 𝑓 ⊆ 𝑆 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin) → ∩ ran 𝑓 ∈ 𝑆))
59	58	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Top ∧ (ran 𝑓 ⊆ 𝑆 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin)) → ∩ ran 𝑓 ∈ 𝑆)
60	45, 48, 49, 57, 59	syl13anc 1372	. . . . . . . . . . . 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 4961	. . . . . . . . . . . 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 4060	. . . . . . . . . 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 4244	. . . . . . . . . 10 ⊢ (∩ ran 𝑓 ⊆ 𝑌 ↔ (𝑌 ∩ ∩ ran 𝑓) = ∩ ran 𝑓)
65	63, 64	sylib 218	. . . . . . . . 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 2844	. . . . . . . 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 3035	. . . . . . 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 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝐴 ∈ 𝑌)
69		simprrr 781	. . . . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈) → 𝐴 ∈ (𝑓‘𝑢))
71	70	ralimi 3089	. . . . . . . . . . 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 5020	. . . . . . . . . . 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 257	. . . . . . . . 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 7000	. . . . . . . . . 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 2846	. . . . . . . 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 4223	. . . . . . 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 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝑋 = ∪ 𝑡)
81		uniiun 5081	. . . . . . . . . . 11 ⊢ ∪ 𝑡 = ∪ 𝑢 ∈ 𝑡 𝑢
82	80, 81	eqtrdi 2796	. . . . . . . . . 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 5729	. . . . . . . . 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 5771	. . . . . . . . 9 ⊢ (∪ 𝑢 ∈ 𝑡 𝑢 × (𝑌 ∩ ∩ ran 𝑓)) = ∪ 𝑢 ∈ 𝑡 (𝑢 × (𝑌 ∩ ∩ ran 𝑓))
85	83, 84	eqtrdi 2796	. . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈) → (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)
87	86	ralimi 3089	. . . . . . . . . . 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 4259	. . . . . . . . . . . . 13 ⊢ (𝑌 ∩ ∩ ran 𝑓) ⊆ ∩ ran 𝑓
90	76	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) = ∩ ran 𝑓)
91		iinss2 5080	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑡 → ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) ⊆ (𝑓‘𝑢))
92	91	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) ⊆ (𝑓‘𝑢))
93	90, 92	eqsstrrd 4048	. . . . . . . . . . . . 13 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → ∩ ran 𝑓 ⊆ (𝑓‘𝑢))
94	89, 93	sstrid 4020	. . . . . . . . . . . 12 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → (𝑌 ∩ ∩ ran 𝑓) ⊆ (𝑓‘𝑢))
95		xpss2 5720	. . . . . . . . . . . 12 ⊢ ((𝑌 ∩ ∩ ran 𝑓) ⊆ (𝑓‘𝑢) → (𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ (𝑢 × (𝑓‘𝑢)))
96		sstr2 4015	. . . . . . . . . . . 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 3173	. . . . . . . . . 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 5068	. . . . . . . . 9 ⊢ (∪ 𝑢 ∈ 𝑡 (𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈 ↔ ∀𝑢 ∈ 𝑡 (𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈)
101	99, 100	sylibr 234	. . . . . . . 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 4047	. . . . . . 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 2833	. . . . . . . . 9 ⊢ (𝑢 = (𝑌 ∩ ∩ ran 𝑓) → (𝐴 ∈ 𝑢 ↔ 𝐴 ∈ (𝑌 ∩ ∩ ran 𝑓)))
104		xpeq2 5721	. . . . . . . . . 10 ⊢ (𝑢 = (𝑌 ∩ ∩ ran 𝑓) → (𝑋 × 𝑢) = (𝑋 × (𝑌 ∩ ∩ ran 𝑓)))
105	104	sseq1d 4040	. . . . . . . . 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 3635	. . . . . . 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 836	. . . . . 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 456	. . . . 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 1932	. . . 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 453	. . 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 3161	. 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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ⟨cop 4654  ∪ cuni 4931  ∩ cint 4970  ∪ ciun 5015  ∩ ciin 5016   × cxp 5698  ran crn 5701   Fn wfn 6568  ⟶wf 6569  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  Topctop 22920  Compccmp 23415   ×_t ctx 23589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-1o 8522  df-2o 8523  df-en 9004  df-dom 9005  df-fin 9007  df-topgen 17503  df-top 22921  df-cmp 23416  df-tx 23591

This theorem is referenced by:  txcmplem1  23670  xkoinjcn  23716  cvmlift2lem12  35282