Theorem txtube 23564

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 2817	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑥, 𝐴⟩ → (𝑦 ∈ (𝑢 × 𝑣) ↔ ⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣)))
3	2	anbi1d 629	. . . . . . 7 ⊢ (𝑦 = ⟨𝑥, 𝐴⟩ → ((𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
4	3	2rexbidv 3217	. . . . . 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 23492	. . . . . . . . 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 582	. . . . . . . 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 479	. . . . . 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 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑋 × {𝐴}) ⊆ 𝑈)
13		id 22	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝑋)
14		txtube.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑌)
15		snidg 4667	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑌 → 𝐴 ∈ {𝐴})
16	14, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
17		opelxpi 5719	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ {𝐴}) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × {𝐴}))
18	13, 16, 17	syl2anr 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × {𝐴}))
19	12, 18	sseldd 3983	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝑥, 𝐴⟩ ∈ 𝑈)
20	4, 10, 19	rspcdva 3612	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
21		opelxp 5718	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ↔ (𝑥 ∈ 𝑢 ∧ 𝐴 ∈ 𝑣))
22	21	anbi1i 622	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ((𝑥 ∈ 𝑢 ∧ 𝐴 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
23		anass 467	. . . . . . . . 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 3091	. . . . . . 7 ⊢ (∃𝑣 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑣 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
26		r19.42v 3188	. . . . . . 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 3091	. . . . 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 3143	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑅 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑆 (𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
31		txtube.x	. . . 4 ⊢ 𝑋 = ∪ 𝑅
32		eleq2 2818	. . . . 5 ⊢ (𝑣 = (𝑓‘𝑢) → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ (𝑓‘𝑢)))
33		xpeq2 5703	. . . . . 6 ⊢ (𝑣 = (𝑓‘𝑢) → (𝑢 × 𝑣) = (𝑢 × (𝑓‘𝑢)))
34	33	sseq1d 4013	. . . . 5 ⊢ (𝑣 = (𝑓‘𝑢) → ((𝑢 × 𝑣) ⊆ 𝑈 ↔ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈))
35	32, 34	anbi12d 630	. . . 4 ⊢ (𝑣 = (𝑓‘𝑢) → ((𝐴 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))
36	31, 35	cmpcovf 23315	. . 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 582	. 2 ⊢ (𝜑 → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = ∪ 𝑡 ∧ ∃𝑓(𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈))))
38		rint0 4997	. . . . . . . . . 10 ⊢ (ran 𝑓 = ∅ → (𝑌 ∩ ∩ ran 𝑓) = 𝑌)
39	38	adantl 480	. . . . . . . . 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 22828	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Top → 𝑌 ∈ 𝑆)
42	6, 41	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
43	42	ad3antrrr 728	. . . . . . . . 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 2829	. . . . . . . 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 728	. . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . 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 6735	. . . . . . . . . . . . . 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 479	. . . . . . . . . . . . 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 483	. . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . 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 4201	. . . . . . . . . . . . . . 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 6728	. . . . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . 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 9370	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ Fin ∧ 𝑓:𝑡–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
56	51, 54, 55	syl2anc 582	. . . . . . . . . . . . . 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 479	. . . . . . . . . . . . 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 22823	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ Top → ((ran 𝑓 ⊆ 𝑆 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin) → ∩ ran 𝑓 ∈ 𝑆))
59	58	imp 405	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Top ∧ (ran 𝑓 ⊆ 𝑆 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin)) → ∩ ran 𝑓 ∈ 𝑆)
60	45, 48, 49, 57, 59	syl13anc 1369	. . . . . . . . . . . 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 4944	. . . . . . . . . . . 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 4033	. . . . . . . . . 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 4217	. . . . . . . . . 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 2829	. . . . . . . 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 3026	. . . . . . 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 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝐴 ∈ 𝑌)
69		simprrr 780	. . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈) → 𝐴 ∈ (𝑓‘𝑢))
71	70	ralimi 3080	. . . . . . . . . . 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 5005	. . . . . . . . . . 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 6981	. . . . . . . . . 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 2831	. . . . . . . 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 4196	. . . . . . 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 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑆 ∧ ∀𝑢 ∈ 𝑡 (𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)))) → 𝑋 = ∪ 𝑡)
81		uniiun 5065	. . . . . . . . . . 11 ⊢ ∪ 𝑡 = ∪ 𝑢 ∈ 𝑡 𝑢
82	80, 81	eqtrdi 2784	. . . . . . . . . 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 5711	. . . . . . . . 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 5753	. . . . . . . . 9 ⊢ (∪ 𝑢 ∈ 𝑡 𝑢 × (𝑌 ∩ ∩ ran 𝑓)) = ∪ 𝑢 ∈ 𝑡 (𝑢 × (𝑌 ∩ ∩ ran 𝑓))
85	83, 84	eqtrdi 2784	. . . . . . . 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 483	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑓‘𝑢) ∧ (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈) → (𝑢 × (𝑓‘𝑢)) ⊆ 𝑈)
87	86	ralimi 3080	. . . . . . . . . . 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 4232	. . . . . . . . . . . . 13 ⊢ (𝑌 ∩ ∩ ran 𝑓) ⊆ ∩ ran 𝑓
90	76	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) = ∩ ran 𝑓)
91		iinss2 5064	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑡 → ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) ⊆ (𝑓‘𝑢))
92	91	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → ∩ 𝑢 ∈ 𝑡 (𝑓‘𝑢) ⊆ (𝑓‘𝑢))
93	90, 92	eqsstrrd 4021	. . . . . . . . . . . . 13 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → ∩ ran 𝑓 ⊆ (𝑓‘𝑢))
94	89, 93	sstrid 3993	. . . . . . . . . . . 12 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → (𝑌 ∩ ∩ ran 𝑓) ⊆ (𝑓‘𝑢))
95		xpss2 5702	. . . . . . . . . . . 12 ⊢ ((𝑌 ∩ ∩ ran 𝑓) ⊆ (𝑓‘𝑢) → (𝑢 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ (𝑢 × (𝑓‘𝑢)))
96		sstr2 3989	. . . . . . . . . . . 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 3164	. . . . . . . . . 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 5052	. . . . . . . . 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 4020	. . . . . . 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 2818	. . . . . . . . 9 ⊢ (𝑢 = (𝑌 ∩ ∩ ran 𝑓) → (𝐴 ∈ 𝑢 ↔ 𝐴 ∈ (𝑌 ∩ ∩ ran 𝑓)))
104		xpeq2 5703	. . . . . . . . . 10 ⊢ (𝑢 = (𝑌 ∩ ∩ ran 𝑓) → (𝑋 × 𝑢) = (𝑋 × (𝑌 ∩ ∩ ran 𝑓)))
105	104	sseq1d 4013	. . . . . . . . 9 ⊢ (𝑢 = (𝑌 ∩ ∩ ran 𝑓) → ((𝑋 × 𝑢) ⊆ 𝑈 ↔ (𝑋 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈))
106	103, 105	anbi12d 630	. . . . . . . 8 ⊢ (𝑢 = (𝑌 ∩ ∩ ran 𝑓) → ((𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈) ↔ (𝐴 ∈ (𝑌 ∩ ∩ ran 𝑓) ∧ (𝑋 × (𝑌 ∩ ∩ ran 𝑓)) ⊆ 𝑈)))
107	106	rspcev 3611	. . . . . . 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 835	. . . . . 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 455	. . . . 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 1928	. . . 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 452	. . 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 3152	. 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 394   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098   ≠ wne 2937  ∀wral 3058  ∃wrex 3067   ∩ cin 3948   ⊆ wss 3949  ∅c0 4326  𝒫 cpw 4606  {csn 4632  ⟨cop 4638  ∪ cuni 4912  ∩ cint 4953  ∪ ciun 5000  ∩ ciin 5001   × cxp 5680  ran crn 5683   Fn wfn 6548  ⟶wf 6549  –onto→wfo 6551  ‘cfv 6553  (class class class)co 7426  Fincfn 8970  Topctop 22815  Compccmp 23310   ×_t ctx 23484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-fin 8974  df-topgen 17432  df-top 22816  df-cmp 23311  df-tx 23486

This theorem is referenced by:  txcmplem1  23565  xkoinjcn  23611  cvmlift2lem12  34957