Theorem lmss 23336

Description: Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)

Hypotheses

Ref	Expression
lmss.1	⊢ 𝐾 = (𝐽 ↾t 𝑌)
lmss.2	⊢ 𝑍 = (ℤ≥‘𝑀)
lmss.3	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lmss.4	⊢ (𝜑 → 𝐽 ∈ Top)
lmss.5	⊢ (𝜑 → 𝑃 ∈ 𝑌)
lmss.6	⊢ (𝜑 → 𝑀 ∈ ℤ)
lmss.7	⊢ (𝜑 → 𝐹:𝑍⟶𝑌)

Assertion

Ref	Expression
lmss	⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹(⇝𝑡‘𝐾)𝑃))

Proof of Theorem lmss

Dummy variables 𝑗 𝑘 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmss.4	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top)
2		toptopon2 22956	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 220	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		lmcl 23335	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ ∪ 𝐽)
5	3, 4	sylan 589	. . . 4 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ ∪ 𝐽)
6		lmfss 23334	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝐹 ⊆ (ℂ × ∪ 𝐽))
7	3, 6	sylan 589	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝐹 ⊆ (ℂ × ∪ 𝐽))
8		rnss 5913	. . . . . 6 ⊢ (𝐹 ⊆ (ℂ × ∪ 𝐽) → ran 𝐹 ⊆ ran (ℂ × ∪ 𝐽))
9	7, 8	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → ran 𝐹 ⊆ ran (ℂ × ∪ 𝐽))
10		rnxpss 6152	. . . . 5 ⊢ ran (ℂ × ∪ 𝐽) ⊆ ∪ 𝐽
11	9, 10	sstrdi 3948	. . . 4 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → ran 𝐹 ⊆ ∪ 𝐽)
12	5, 11	jca 519	. . 3 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽))
13	12	ex 416	. 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 → (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)))
14		lmss.1	. . . . . . 7 ⊢ 𝐾 = (𝐽 ↾t 𝑌)
15		lmss.3	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
16		resttopon2 23206	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑌 ∈ 𝑉) → (𝐽 ↾t 𝑌) ∈ (TopOn‘(𝑌 ∩ ∪ 𝐽)))
17	3, 15, 16	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾t 𝑌) ∈ (TopOn‘(𝑌 ∩ ∪ 𝐽)))
18	14, 17	eqeltrid 2865	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(𝑌 ∩ ∪ 𝐽)))
19		lmcl 23335	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘(𝑌 ∩ ∪ 𝐽)) ∧ 𝐹(⇝𝑡‘𝐾)𝑃) → 𝑃 ∈ (𝑌 ∩ ∪ 𝐽))
20	18, 19	sylan 589	. . . . 5 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐾)𝑃) → 𝑃 ∈ (𝑌 ∩ ∪ 𝐽))
21	20	elin2d 4157	. . . 4 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐾)𝑃) → 𝑃 ∈ ∪ 𝐽)
22		lmfss 23334	. . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘(𝑌 ∩ ∪ 𝐽)) ∧ 𝐹(⇝𝑡‘𝐾)𝑃) → 𝐹 ⊆ (ℂ × (𝑌 ∩ ∪ 𝐽)))
23	18, 22	sylan 589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐾)𝑃) → 𝐹 ⊆ (ℂ × (𝑌 ∩ ∪ 𝐽)))
24		rnss 5913	. . . . . . 7 ⊢ (𝐹 ⊆ (ℂ × (𝑌 ∩ ∪ 𝐽)) → ran 𝐹 ⊆ ran (ℂ × (𝑌 ∩ ∪ 𝐽)))
25	23, 24	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐾)𝑃) → ran 𝐹 ⊆ ran (ℂ × (𝑌 ∩ ∪ 𝐽)))
26		rnxpss 6152	. . . . . 6 ⊢ ran (ℂ × (𝑌 ∩ ∪ 𝐽)) ⊆ (𝑌 ∩ ∪ 𝐽)
27	25, 26	sstrdi 3948	. . . . 5 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐾)𝑃) → ran 𝐹 ⊆ (𝑌 ∩ ∪ 𝐽))
28		inss2 4189	. . . . 5 ⊢ (𝑌 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
29	27, 28	sstrdi 3948	. . . 4 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐾)𝑃) → ran 𝐹 ⊆ ∪ 𝐽)
30	21, 29	jca 519	. . 3 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐾)𝑃) → (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽))
31	30	ex 416	. 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐾)𝑃 → (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)))
32		simprl 780	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → 𝑃 ∈ ∪ 𝐽)
33		lmss.5	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ 𝑌)
34	33	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → 𝑃 ∈ 𝑌)
35	34, 32	elind 4152	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → 𝑃 ∈ (𝑌 ∩ ∪ 𝐽))
36	32, 35	2thd 267	. . . . 5 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → (𝑃 ∈ ∪ 𝐽 ↔ 𝑃 ∈ (𝑌 ∩ ∪ 𝐽)))
37	14	eleq2i 2853	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐾 ↔ 𝑣 ∈ (𝐽 ↾t 𝑌))
38	1	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → 𝐽 ∈ Top)
39	15	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → 𝑌 ∈ 𝑉)
40		elrest 17437	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ 𝑉) → (𝑣 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑣 = (𝑢 ∩ 𝑌)))
41	38, 39, 40	syl2anc 593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → (𝑣 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑣 = (𝑢 ∩ 𝑌)))
42	41	biimpa 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑣 ∈ (𝐽 ↾t 𝑌)) → ∃𝑢 ∈ 𝐽 𝑣 = (𝑢 ∩ 𝑌))
43	37, 42	sylan2b 603	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑣 ∈ 𝐾) → ∃𝑢 ∈ 𝐽 𝑣 = (𝑢 ∩ 𝑌))
44		r19.29r 3125	. . . . . . . . . 10 ⊢ ((∃𝑢 ∈ 𝐽 𝑣 = (𝑢 ∩ 𝑌) ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)) → ∃𝑢 ∈ 𝐽 (𝑣 = (𝑢 ∩ 𝑌) ∧ (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
45	34	biantrud 539	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → (𝑃 ∈ 𝑢 ↔ (𝑃 ∈ 𝑢 ∧ 𝑃 ∈ 𝑌)))
46		elin 3920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (𝑢 ∩ 𝑌) ↔ (𝑃 ∈ 𝑢 ∧ 𝑃 ∈ 𝑌))
47	45, 46	bitr4di 291	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → (𝑃 ∈ 𝑢 ↔ 𝑃 ∈ (𝑢 ∩ 𝑌)))
48		lmss.2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑍 = (ℤ≥‘𝑀)
49	48	uztrn2 12853	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
50		lmss.7	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐹:𝑍⟶𝑌)
51	50	adantr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → 𝐹:𝑍⟶𝑌)
52	51	ffvelcdmda 7059	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ 𝑌)
53	52	biantrud 539	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) ∈ 𝑢 ↔ ((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐹‘𝑘) ∈ 𝑌)))
54		elin 3920	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌) ↔ ((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐹‘𝑘) ∈ 𝑌))
55	53, 54	bitr4di 291	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) ∈ 𝑢 ↔ (𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌)))
56	49, 55	sylan2 602	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗))) → ((𝐹‘𝑘) ∈ 𝑢 ↔ (𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌)))
57	56	anassrs 471	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐹‘𝑘) ∈ 𝑢 ↔ (𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌)))
58	57	ralbidva 3182	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌)))
59	58	rexbidva 3183	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌)))
60	47, 59	imbi12d 346	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → ((𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ↔ (𝑃 ∈ (𝑢 ∩ 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌))))
61	60	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑢 ∈ 𝐽) → ((𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ↔ (𝑃 ∈ (𝑢 ∩ 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌))))
62	61	biimpd 231	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑢 ∈ 𝐽) → ((𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) → (𝑃 ∈ (𝑢 ∩ 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌))))
63		eleq2 2850	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑢 ∩ 𝑌) → (𝑃 ∈ 𝑣 ↔ 𝑃 ∈ (𝑢 ∩ 𝑌)))
64		eleq2 2850	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (𝑢 ∩ 𝑌) → ((𝐹‘𝑘) ∈ 𝑣 ↔ (𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌)))
65	64	rexralbidv 3227	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑢 ∩ 𝑌) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌)))
66	63, 65	imbi12d 346	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑢 ∩ 𝑌) → ((𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣) ↔ (𝑃 ∈ (𝑢 ∩ 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌))))
67	66	imbi2d 342	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑢 ∩ 𝑌) → (((𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) → (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣)) ↔ ((𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) → (𝑃 ∈ (𝑢 ∩ 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌)))))
68	62, 67	syl5ibrcom 249	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑢 ∈ 𝐽) → (𝑣 = (𝑢 ∩ 𝑌) → ((𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) → (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣))))
69	68	impd 414	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑢 ∈ 𝐽) → ((𝑣 = (𝑢 ∩ 𝑌) ∧ (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)) → (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣)))
70	69	rexlimdva 3162	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → (∃𝑢 ∈ 𝐽 (𝑣 = (𝑢 ∩ 𝑌) ∧ (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)) → (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣)))
71	44, 70	syl5 34	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → ((∃𝑢 ∈ 𝐽 𝑣 = (𝑢 ∩ 𝑌) ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)) → (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣)))
72	71	expdimp 456	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ ∃𝑢 ∈ 𝐽 𝑣 = (𝑢 ∩ 𝑌)) → (∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) → (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣)))
73	43, 72	syldan 600	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑣 ∈ 𝐾) → (∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) → (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣)))
74	73	ralrimdva 3161	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → (∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) → ∀𝑣 ∈ 𝐾 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣)))
75	38	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑢 ∈ 𝐽) → 𝐽 ∈ Top)
76	39	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑢 ∈ 𝐽) → 𝑌 ∈ 𝑉)
77		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑢 ∈ 𝐽) → 𝑢 ∈ 𝐽)
78		elrestr 17438	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ 𝑉 ∧ 𝑢 ∈ 𝐽) → (𝑢 ∩ 𝑌) ∈ (𝐽 ↾t 𝑌))
79	75, 76, 77, 78	syl3anc 1389	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑢 ∈ 𝐽) → (𝑢 ∩ 𝑌) ∈ (𝐽 ↾t 𝑌))
80	79, 14	eleqtrrdi 2872	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑢 ∈ 𝐽) → (𝑢 ∩ 𝑌) ∈ 𝐾)
81	66	rspcv 3577	. . . . . . . . 9 ⊢ ((𝑢 ∩ 𝑌) ∈ 𝐾 → (∀𝑣 ∈ 𝐾 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣) → (𝑃 ∈ (𝑢 ∩ 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌))))
82	80, 81	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑢 ∈ 𝐽) → (∀𝑣 ∈ 𝐾 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣) → (𝑃 ∈ (𝑢 ∩ 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ (𝑢 ∩ 𝑌))))
83	82, 61	sylibrd 261	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑢 ∈ 𝐽) → (∀𝑣 ∈ 𝐾 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣) → (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
84	83	ralrimdva 3161	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → (∀𝑣 ∈ 𝐾 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣) → ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
85	74, 84	impbid 214	. . . . 5 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → (∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ↔ ∀𝑣 ∈ 𝐾 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣)))
86	36, 85	anbi12d 641	. . . 4 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → ((𝑃 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)) ↔ (𝑃 ∈ (𝑌 ∩ ∪ 𝐽) ∧ ∀𝑣 ∈ 𝐾 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣))))
87	38, 2	sylib 220	. . . . 5 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
88		lmss.6	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
89	88	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → 𝑀 ∈ ℤ)
90	51	ffnd 6686	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → 𝐹 Fn 𝑍)
91		simprr 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → ran 𝐹 ⊆ ∪ 𝐽)
92		df-f 6519	. . . . . 6 ⊢ (𝐹:𝑍⟶∪ 𝐽 ↔ (𝐹 Fn 𝑍 ∧ ran 𝐹 ⊆ ∪ 𝐽))
93	90, 91, 92	sylanbrc 592	. . . . 5 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → 𝐹:𝑍⟶∪ 𝐽)
94		eqidd 2762	. . . . 5 ⊢ (((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
95	87, 48, 89, 93, 94	lmbrf 23298	. . . 4 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝑃 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))))
96	18	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → 𝐾 ∈ (TopOn‘(𝑌 ∩ ∪ 𝐽)))
97	51	frnd 6694	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → ran 𝐹 ⊆ 𝑌)
98	97, 91	ssind 4192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → ran 𝐹 ⊆ (𝑌 ∩ ∪ 𝐽))
99		df-f 6519	. . . . . 6 ⊢ (𝐹:𝑍⟶(𝑌 ∩ ∪ 𝐽) ↔ (𝐹 Fn 𝑍 ∧ ran 𝐹 ⊆ (𝑌 ∩ ∪ 𝐽)))
100	90, 98, 99	sylanbrc 592	. . . . 5 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → 𝐹:𝑍⟶(𝑌 ∩ ∪ 𝐽))
101	96, 48, 89, 100, 94	lmbrf 23298	. . . 4 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → (𝐹(⇝𝑡‘𝐾)𝑃 ↔ (𝑃 ∈ (𝑌 ∩ ∪ 𝐽) ∧ ∀𝑣 ∈ 𝐾 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑣))))
102	86, 95, 101	3bitr4d 313	. . 3 ⊢ ((𝜑 ∧ (𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽)) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹(⇝𝑡‘𝐾)𝑃))
103	102	ex 416	. 2 ⊢ (𝜑 → ((𝑃 ∈ ∪ 𝐽 ∧ ran 𝐹 ⊆ ∪ 𝐽) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹(⇝𝑡‘𝐾)𝑃)))
104	13, 31, 103	pm5.21ndd 381	1 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹(⇝𝑡‘𝐾)𝑃))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   ∩ cin 3903   ⊆ wss 3904  ∪ cuni 4864   class class class wbr 5099   × cxp 5643  ran crn 5646   Fn wfn 6510  ⟶wf 6511  ‘cfv 6515  (class class class)co 7390  ℂcc 11066  ℤcz 12563  ℤ_≥cuz 12834   ↾_t crest 17430  Topctop 22931  TopOnctopon 22948  ⇝_𝑡clm 23264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7712  ax-cnex 11124  ax-resscn 11125  ax-pre-lttri 11142  ax-pre-lttrn 11143

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-er 8671  df-pm 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fi 9352  df-pnf 11213  df-mnf 11214  df-xr 11215  df-ltxr 11216  df-le 11217  df-neg 11412  df-z 12564  df-uz 12835  df-rest 17432  df-topgen 17453  df-top 22932  df-topon 22949  df-bases 22984  df-lm 23267

This theorem is referenced by:  1stckgen  23592  minvecolem4b  31025  minvecolem4  31027  hhsscms  31425  lmlim  34203  climreeq  46142  xlimclim  46351