Theorem cvmsss2 31256

Description: An open subset of an evenly covered set is evenly covered. (Contributed by Mario Carneiro, 7-Jul-2015.)

Hypothesis

Ref	Expression
cvmcov.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})

Assertion

Ref	Expression
cvmsss2	⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → ((𝑆‘𝑈) ≠ ∅ → (𝑆‘𝑉) ≠ ∅))

Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣   𝑘,𝐽,𝑠,𝑢,𝑣   𝑈,𝑘,𝑠,𝑢,𝑣   𝑘,𝑉,𝑠,𝑢,𝑣

Allowed substitution hints:   𝑆(𝑣,𝑢,𝑘,𝑠)

Proof of Theorem cvmsss2

Dummy variables 𝑎 𝑏 𝑡 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. 2 ⊢ ((𝑆‘𝑈) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆‘𝑈))
2		simpl2 1065	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑉 ∈ 𝐽)
3		simpl1 1064	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
4		cvmtop1 31242	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
5	3, 4	syl 17	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝐶 ∈ Top)
6	5	adantr 481	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → 𝐶 ∈ Top)
7		cvmcov.1	. . . . . . . . . . . . 13 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})
8	7	cvmsss 31249	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑆‘𝑈) → 𝑥 ⊆ 𝐶)
9	8	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑥 ⊆ 𝐶)
10	9	sselda 3603	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐶)
11		cvmcn 31244	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
12	3, 11	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝐹 ∈ (𝐶 Cn 𝐽))
13		cnima 21069	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝑉 ∈ 𝐽) → (◡𝐹 “ 𝑉) ∈ 𝐶)
14	12, 2, 13	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (◡𝐹 “ 𝑉) ∈ 𝐶)
15	14	adantr 481	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → (◡𝐹 “ 𝑉) ∈ 𝐶)
16		inopn 20704	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Top ∧ 𝑦 ∈ 𝐶 ∧ (◡𝐹 “ 𝑉) ∈ 𝐶) → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝐶)
17	6, 10, 15, 16	syl3anc 1326	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝐶)
18		eqid 2622	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))
19	17, 18	fmptd 6385	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))):𝑥⟶𝐶)
20		frn 6053	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))):𝑥⟶𝐶 → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝐶)
21	19, 20	syl 17	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝐶)
22	7	cvmsn0 31250	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑆‘𝑈) → 𝑥 ≠ ∅)
23	22	adantl 482	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑥 ≠ ∅)
24		dmmptg 5632	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ V → dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = 𝑥)
25		inex1g 4801	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑥 → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ V)
26	24, 25	mprg 2926	. . . . . . . . . . 11 ⊢ dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = 𝑥
27	26	eqeq1i 2627	. . . . . . . . . 10 ⊢ (dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅ ↔ 𝑥 = ∅)
28		dm0rn0 5342	. . . . . . . . . 10 ⊢ (dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅ ↔ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅)
29	27, 28	bitr3i 266	. . . . . . . . 9 ⊢ (𝑥 = ∅ ↔ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅)
30	29	necon3bii 2846	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ≠ ∅)
31	23, 30	sylib 208	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ≠ ∅)
32	21, 31	jca 554	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝐶 ∧ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ≠ ∅))
33		inss2 3834	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ (◡𝐹 “ 𝑉)) ⊆ (◡𝐹 “ 𝑉)
34		elpw2g 4827	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ 𝑉) ∈ 𝐶 → ((𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝒫 (◡𝐹 “ 𝑉) ↔ (𝑦 ∩ (◡𝐹 “ 𝑉)) ⊆ (◡𝐹 “ 𝑉)))
35	15, 34	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝒫 (◡𝐹 “ 𝑉) ↔ (𝑦 ∩ (◡𝐹 “ 𝑉)) ⊆ (◡𝐹 “ 𝑉)))
36	33, 35	mpbiri 248	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝒫 (◡𝐹 “ 𝑉))
37	36, 18	fmptd 6385	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))):𝑥⟶𝒫 (◡𝐹 “ 𝑉))
38		frn 6053	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))):𝑥⟶𝒫 (◡𝐹 “ 𝑉) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝒫 (◡𝐹 “ 𝑉))
39	37, 38	syl 17	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝒫 (◡𝐹 “ 𝑉))
40		sspwuni 4611	. . . . . . . . 9 ⊢ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝒫 (◡𝐹 “ 𝑉) ↔ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ (◡𝐹 “ 𝑉))
41	39, 40	sylib 208	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ (◡𝐹 “ 𝑉))
42		simpl3 1066	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑉 ⊆ 𝑈)
43		imass2 5501	. . . . . . . . . . . . . 14 ⊢ (𝑉 ⊆ 𝑈 → (◡𝐹 “ 𝑉) ⊆ (◡𝐹 “ 𝑈))
44	42, 43	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (◡𝐹 “ 𝑉) ⊆ (◡𝐹 “ 𝑈))
45	7	cvmsuni 31251	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑆‘𝑈) → ∪ 𝑥 = (◡𝐹 “ 𝑈))
46	45	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∪ 𝑥 = (◡𝐹 “ 𝑈))
47	44, 46	sseqtr4d 3642	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (◡𝐹 “ 𝑉) ⊆ ∪ 𝑥)
48	47	sselda 3603	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) → 𝑧 ∈ ∪ 𝑥)
49		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉))
50		ineq1 3807	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑡 → (𝑦 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉)))
51	50	eqeq2d 2632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑡 → ((𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)) ↔ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉))))
52	51	rspcev 3309	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ 𝑥 ∧ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉))) → ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)))
53	49, 52	mpan2 707	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝑥 → ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)))
54	53	ad2antrl 764	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) ∧ (𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡)) → ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)))
55		vex 3203	. . . . . . . . . . . . . . . . 17 ⊢ 𝑡 ∈ V
56	55	inex1 4799	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ V
57	18	elrnmpt 5372	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ V → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ↔ ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉))))
58	56, 57	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ↔ ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)))
59	54, 58	sylibr 224	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) ∧ (𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡)) → (𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))))
60		simprr 796	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) ∧ (𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡)) → 𝑧 ∈ 𝑡)
61		simplr 792	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) ∧ (𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡)) → 𝑧 ∈ (◡𝐹 “ 𝑉))
62	60, 61	elind 3798	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) ∧ (𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡)) → 𝑧 ∈ (𝑡 ∩ (◡𝐹 “ 𝑉)))
63		eleq2 2690	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑡 ∩ (◡𝐹 “ 𝑉))))
64	63	rspcev 3309	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∧ 𝑧 ∈ (𝑡 ∩ (◡𝐹 “ 𝑉))) → ∃𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))𝑧 ∈ 𝑤)
65	59, 62, 64	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) ∧ (𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡)) → ∃𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))𝑧 ∈ 𝑤)
66	65	rexlimdvaa 3032	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) → (∃𝑡 ∈ 𝑥 𝑧 ∈ 𝑡 → ∃𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))𝑧 ∈ 𝑤))
67		eluni2 4440	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑥 ↔ ∃𝑡 ∈ 𝑥 𝑧 ∈ 𝑡)
68		eluni2 4440	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ↔ ∃𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))𝑧 ∈ 𝑤)
69	66, 67, 68	3imtr4g 285	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) → (𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))))
70	48, 69	mpd 15	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) → 𝑧 ∈ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))))
71	70	ex 450	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (𝑧 ∈ (◡𝐹 “ 𝑉) → 𝑧 ∈ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))))
72	71	ssrdv 3609	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (◡𝐹 “ 𝑉) ⊆ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))))
73	41, 72	eqssd 3620	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (◡𝐹 “ 𝑉))
74		eldifsn 4317	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))}) ↔ (𝑧 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∧ 𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉))))
75		vex 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
76	18	elrnmpt 5372	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ↔ ∃𝑦 ∈ 𝑥 𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉))))
77	75, 76	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ↔ ∃𝑦 ∈ 𝑥 𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)))
78	50	equcoms 1947	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑦 → (𝑦 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉)))
79	78	necon3ai 2819	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∩ (◡𝐹 “ 𝑉)) ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) → ¬ 𝑡 = 𝑦)
80		simpllr 799	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ (𝑆‘𝑈))
81		simplr 792	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑡 ∈ 𝑥)
82		simpr 477	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
83	7	cvmsdisj 31252	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (𝑆‘𝑈) ∧ 𝑡 ∈ 𝑥 ∧ 𝑦 ∈ 𝑥) → (𝑡 = 𝑦 ∨ (𝑡 ∩ 𝑦) = ∅))
84	80, 81, 82, 83	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝑡 = 𝑦 ∨ (𝑡 ∩ 𝑦) = ∅))
85	84	ord 392	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (¬ 𝑡 = 𝑦 → (𝑡 ∩ 𝑦) = ∅))
86		inss1 3833	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) ⊆ (𝑡 ∩ 𝑦)
87		sseq0 3975	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) ⊆ (𝑡 ∩ 𝑦) ∧ (𝑡 ∩ 𝑦) = ∅) → ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅)
88	86, 87	mpan 706	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∩ 𝑦) = ∅ → ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅)
89	79, 85, 88	syl56 36	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∩ (◡𝐹 “ 𝑉)) ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅))
90		neeq1 2856	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → (𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) ↔ (𝑦 ∩ (◡𝐹 “ 𝑉)) ≠ (𝑡 ∩ (◡𝐹 “ 𝑉))))
91		ineq2 3808	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ (𝑦 ∩ (◡𝐹 “ 𝑉))))
92		inindir 3831	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ (𝑦 ∩ (◡𝐹 “ 𝑉)))
93	91, 92	syl6eqr 2674	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)))
94	93	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → (((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅ ↔ ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅))
95	90, 94	imbi12d 334	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → ((𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅) ↔ ((𝑦 ∩ (◡𝐹 “ 𝑉)) ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅)))
96	89, 95	syl5ibrcom 237	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → (𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅)))
97	96	rexlimdva 3031	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (∃𝑦 ∈ 𝑥 𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → (𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅)))
98	77, 97	syl5bi 232	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝑧 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) → (𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅)))
99	98	impd 447	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝑧 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∧ 𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉))) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅))
100	74, 99	syl5bi 232	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))}) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅))
101	100	ralrimiv 2965	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))})((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅)
102		inss1 3833	. . . . . . . . . . . . 13 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ 𝑡
103		resabs1 5427	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ 𝑡 → ((𝐹 ↾ 𝑡) ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) = (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))))
104	102, 103	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ 𝑡) ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) = (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉)))
105	7	cvmshmeo 31253	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝑆‘𝑈) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡) ∈ ((𝐶 ↾t 𝑡)Homeo(𝐽 ↾t 𝑈)))
106	105	adantll 750	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡) ∈ ((𝐶 ↾t 𝑡)Homeo(𝐽 ↾t 𝑈)))
107	5	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝐶 ∈ Top)
108	9	sselda 3603	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 ∈ 𝐶)
109		elssuni 4467	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝐶 → 𝑡 ⊆ ∪ 𝐶)
110	108, 109	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 ⊆ ∪ 𝐶)
111		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝐶 = ∪ 𝐶
112	111	restuni 20966	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ Top ∧ 𝑡 ⊆ ∪ 𝐶) → 𝑡 = ∪ (𝐶 ↾t 𝑡))
113	107, 110, 112	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 = ∪ (𝐶 ↾t 𝑡))
114	102, 113	syl5sseq 3653	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ ∪ (𝐶 ↾t 𝑡))
115		eqid 2622	. . . . . . . . . . . . . 14 ⊢ ∪ (𝐶 ↾t 𝑡) = ∪ (𝐶 ↾t 𝑡)
116	115	hmeores 21574	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ 𝑡) ∈ ((𝐶 ↾t 𝑡)Homeo(𝐽 ↾t 𝑈)) ∧ (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ ∪ (𝐶 ↾t 𝑡)) → ((𝐹 ↾ 𝑡) ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) ∈ (((𝐶 ↾t 𝑡) ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo((𝐽 ↾t 𝑈) ↾t ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉))))))
117	106, 114, 116	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐹 ↾ 𝑡) ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) ∈ (((𝐶 ↾t 𝑡) ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo((𝐽 ↾t 𝑈) ↾t ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉))))))
118	104, 117	syl5eqelr 2706	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) ∈ (((𝐶 ↾t 𝑡) ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo((𝐽 ↾t 𝑈) ↾t ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉))))))
119	102	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ 𝑡)
120		simpr 477	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 ∈ 𝑥)
121		restabs 20969	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ Top ∧ (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ 𝑡 ∧ 𝑡 ∈ 𝑥) → ((𝐶 ↾t 𝑡) ↾t (𝑡 ∩ (◡𝐹 “ 𝑉))) = (𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉))))
122	107, 119, 120, 121	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐶 ↾t 𝑡) ↾t (𝑡 ∩ (◡𝐹 “ 𝑉))) = (𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉))))
123		incom 3805	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) = ((◡𝐹 “ 𝑉) ∩ 𝑡)
124		cnvresima 5623	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝐹 ↾ 𝑡) “ 𝑉) = ((◡𝐹 “ 𝑉) ∩ 𝑡)
125	123, 124	eqtr4i 2647	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (◡(𝐹 ↾ 𝑡) “ 𝑉)
126	125	imaeq2i 5464	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉))) = ((𝐹 ↾ 𝑡) “ (◡(𝐹 ↾ 𝑡) “ 𝑉))
127	3	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝐹 ∈ (𝐶 CovMap 𝐽))
128		simplr 792	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑥 ∈ (𝑆‘𝑈))
129	7	cvmsf1o 31254	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥 ∈ (𝑆‘𝑈) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑈)
130	127, 128, 120, 129	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑈)
131		f1ofo 6144	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑈 → (𝐹 ↾ 𝑡):𝑡–onto→𝑈)
132	130, 131	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡):𝑡–onto→𝑈)
133	42	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑉 ⊆ 𝑈)
134		foimacnv 6154	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ↾ 𝑡):𝑡–onto→𝑈 ∧ 𝑉 ⊆ 𝑈) → ((𝐹 ↾ 𝑡) “ (◡(𝐹 ↾ 𝑡) “ 𝑉)) = 𝑉)
135	132, 133, 134	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐹 ↾ 𝑡) “ (◡(𝐹 ↾ 𝑡) “ 𝑉)) = 𝑉)
136	126, 135	syl5eq 2668	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉))) = 𝑉)
137	136	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐽 ↾t 𝑈) ↾t ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉)))) = ((𝐽 ↾t 𝑈) ↾t 𝑉))
138		cvmtop2 31243	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
139	3, 138	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝐽 ∈ Top)
140	7	cvmsrcl 31246	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽)
141	140	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑈 ∈ 𝐽)
142		restabs 20969	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑉 ⊆ 𝑈 ∧ 𝑈 ∈ 𝐽) → ((𝐽 ↾t 𝑈) ↾t 𝑉) = (𝐽 ↾t 𝑉))
143	139, 42, 141, 142	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ((𝐽 ↾t 𝑈) ↾t 𝑉) = (𝐽 ↾t 𝑉))
144	143	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐽 ↾t 𝑈) ↾t 𝑉) = (𝐽 ↾t 𝑉))
145	137, 144	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐽 ↾t 𝑈) ↾t ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉)))) = (𝐽 ↾t 𝑉))
146	122, 145	oveq12d 6668	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (((𝐶 ↾t 𝑡) ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo((𝐽 ↾t 𝑈) ↾t ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉))))) = ((𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo(𝐽 ↾t 𝑉)))
147	118, 146	eleqtrd 2703	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) ∈ ((𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo(𝐽 ↾t 𝑉)))
148	101, 147	jca 554	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))})((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅ ∧ (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) ∈ ((𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo(𝐽 ↾t 𝑉))))
149	148	ralrimiva 2966	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∀𝑡 ∈ 𝑥 (∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))})((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅ ∧ (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) ∈ ((𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo(𝐽 ↾t 𝑉))))
150	56	rgenw 2924	. . . . . . . . 9 ⊢ ∀𝑡 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ V
151	50	cbvmptv 4750	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (𝑡 ∈ 𝑥 ↦ (𝑡 ∩ (◡𝐹 “ 𝑉)))
152		sneq 4187	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → {𝑤} = {(𝑡 ∩ (◡𝐹 “ 𝑉))})
153	152	difeq2d 3728	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {𝑤}) = (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))}))
154		ineq1 3807	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝑤 ∩ 𝑧) = ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧))
155	154	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝑤 ∩ 𝑧) = ∅ ↔ ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅))
156	153, 155	raleqbidv 3152	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {𝑤})(𝑤 ∩ 𝑧) = ∅ ↔ ∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))})((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅))
157		reseq2 5391	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝐹 ↾ 𝑤) = (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))))
158		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝐶 ↾t 𝑤) = (𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉))))
159	158	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝐶 ↾t 𝑤)Homeo(𝐽 ↾t 𝑉)) = ((𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo(𝐽 ↾t 𝑉)))
160	157, 159	eleq12d 2695	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝐹 ↾ 𝑤) ∈ ((𝐶 ↾t 𝑤)Homeo(𝐽 ↾t 𝑉)) ↔ (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) ∈ ((𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo(𝐽 ↾t 𝑉))))
161	156, 160	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {𝑤})(𝑤 ∩ 𝑧) = ∅ ∧ (𝐹 ↾ 𝑤) ∈ ((𝐶 ↾t 𝑤)Homeo(𝐽 ↾t 𝑉))) ↔ (∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))})((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅ ∧ (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) ∈ ((𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo(𝐽 ↾t 𝑉)))))
162	151, 161	ralrnmpt 6368	. . . . . . . . 9 ⊢ (∀𝑡 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ V → (∀𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))(∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {𝑤})(𝑤 ∩ 𝑧) = ∅ ∧ (𝐹 ↾ 𝑤) ∈ ((𝐶 ↾t 𝑤)Homeo(𝐽 ↾t 𝑉))) ↔ ∀𝑡 ∈ 𝑥 (∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))})((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅ ∧ (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) ∈ ((𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo(𝐽 ↾t 𝑉)))))
163	150, 162	ax-mp 5	. . . . . . . 8 ⊢ (∀𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))(∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {𝑤})(𝑤 ∩ 𝑧) = ∅ ∧ (𝐹 ↾ 𝑤) ∈ ((𝐶 ↾t 𝑤)Homeo(𝐽 ↾t 𝑉))) ↔ ∀𝑡 ∈ 𝑥 (∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))})((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅ ∧ (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) ∈ ((𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo(𝐽 ↾t 𝑉))))
164	149, 163	sylibr 224	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∀𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))(∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {𝑤})(𝑤 ∩ 𝑧) = ∅ ∧ (𝐹 ↾ 𝑤) ∈ ((𝐶 ↾t 𝑤)Homeo(𝐽 ↾t 𝑉))))
165	73, 164	jca 554	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (◡𝐹 “ 𝑉) ∧ ∀𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))(∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {𝑤})(𝑤 ∩ 𝑧) = ∅ ∧ (𝐹 ↾ 𝑤) ∈ ((𝐶 ↾t 𝑤)Homeo(𝐽 ↾t 𝑉)))))
166	7	cvmscbv 31240	. . . . . . . 8 ⊢ 𝑆 = (𝑎 ∈ 𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑏 = (◡𝐹 “ 𝑎) ∧ ∀𝑤 ∈ 𝑏 (∀𝑧 ∈ (𝑏 ∖ {𝑤})(𝑤 ∩ 𝑧) = ∅ ∧ (𝐹 ↾ 𝑤) ∈ ((𝐶 ↾t 𝑤)Homeo(𝐽 ↾t 𝑎))))})
167	166	cvmsval 31248	. . . . . . 7 ⊢ (𝐶 ∈ Top → (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∈ (𝑆‘𝑉) ↔ (𝑉 ∈ 𝐽 ∧ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝐶 ∧ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ≠ ∅) ∧ (∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (◡𝐹 “ 𝑉) ∧ ∀𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))(∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {𝑤})(𝑤 ∩ 𝑧) = ∅ ∧ (𝐹 ↾ 𝑤) ∈ ((𝐶 ↾t 𝑤)Homeo(𝐽 ↾t 𝑉)))))))
168	5, 167	syl 17	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∈ (𝑆‘𝑉) ↔ (𝑉 ∈ 𝐽 ∧ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝐶 ∧ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ≠ ∅) ∧ (∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (◡𝐹 “ 𝑉) ∧ ∀𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))(∀𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {𝑤})(𝑤 ∩ 𝑧) = ∅ ∧ (𝐹 ↾ 𝑤) ∈ ((𝐶 ↾t 𝑤)Homeo(𝐽 ↾t 𝑉)))))))
169	2, 32, 165, 168	mpbir3and 1245	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∈ (𝑆‘𝑉))
170		ne0i 3921	. . . . 5 ⊢ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∈ (𝑆‘𝑉) → (𝑆‘𝑉) ≠ ∅)
171	169, 170	syl 17	. . . 4 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (𝑆‘𝑉) ≠ ∅)
172	171	ex 450	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → (𝑥 ∈ (𝑆‘𝑈) → (𝑆‘𝑉) ≠ ∅))
173	172	exlimdv 1861	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → (∃𝑥 𝑥 ∈ (𝑆‘𝑈) → (𝑆‘𝑉) ≠ ∅))
174	1, 173	syl5bi 232	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → ((𝑆‘𝑈) ≠ ∅ → (𝑆‘𝑉) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↾_t crest 16081  Topctop 20698   Cn ccn 21028  Homeochmeo 21556   CovMap ccvm 31237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-hmeo 21558  df-cvm 31238

This theorem is referenced by:  cvmcov2  31257