Theorem cvmsss2 30510

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 3890	. 2 ⊢ ((𝑆‘𝑈) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆‘𝑈))
2		simpl2 1058	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑉 ∈ 𝐽)
3		simpl1 1057	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
4		cvmtop1 30496	. . . . . . . . . . . 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 480	. . . . . . . . . 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 30503	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑆‘𝑈) → 𝑥 ⊆ 𝐶)
9	8	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑥 ⊆ 𝐶)
10	9	sselda 3568	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐶)
11		cvmcn 30498	. . . . . . . . . . . . 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 20879	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝑉 ∈ 𝐽) → (◡𝐹 “ 𝑉) ∈ 𝐶)
14	12, 2, 13	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (◡𝐹 “ 𝑉) ∈ 𝐶)
15	14	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → (◡𝐹 “ 𝑉) ∈ 𝐶)
16		inopn 20529	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Top ∧ 𝑦 ∈ 𝐶 ∧ (◡𝐹 “ 𝑉) ∈ 𝐶) → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝐶)
17	6, 10, 15, 16	syl3anc 1318	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝐶)
18		eqid 2610	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))
19	17, 18	fmptd 6292	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))):𝑥⟶𝐶)
20		frn 5966	. . . . . . . 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 30504	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑆‘𝑈) → 𝑥 ≠ ∅)
23	22	adantl 481	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑥 ≠ ∅)
24		dmmptg 5549	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ V → dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = 𝑥)
25		inex1g 4729	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑥 → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ V)
26	24, 25	mprg 2910	. . . . . . . . . . 11 ⊢ dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = 𝑥
27	26	eqeq1i 2615	. . . . . . . . . 10 ⊢ (dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅ ↔ 𝑥 = ∅)
28		dm0rn0 5263	. . . . . . . . . 10 ⊢ (dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅ ↔ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅)
29	27, 28	bitr3i 265	. . . . . . . . 9 ⊢ (𝑥 = ∅ ↔ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅)
30	29	necon3bii 2834	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ≠ ∅)
31	23, 30	sylib 207	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ≠ ∅)
32	21, 31	jca 553	. . . . . 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 3796	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ (◡𝐹 “ 𝑉)) ⊆ (◡𝐹 “ 𝑉)
34		elpw2g 4754	. . . . . . . . . . . . 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 247	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝒫 (◡𝐹 “ 𝑉))
37	36, 18	fmptd 6292	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))):𝑥⟶𝒫 (◡𝐹 “ 𝑉))
38		frn 5966	. . . . . . . . . 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 4547	. . . . . . . . 9 ⊢ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝒫 (◡𝐹 “ 𝑉) ↔ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ (◡𝐹 “ 𝑉))
41	39, 40	sylib 207	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ (◡𝐹 “ 𝑉))
42		simpl3 1059	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑉 ⊆ 𝑈)
43		imass2 5420	. . . . . . . . . . . . . 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 30505	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑆‘𝑈) → ∪ 𝑥 = (◡𝐹 “ 𝑈))
46	45	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∪ 𝑥 = (◡𝐹 “ 𝑈))
47	44, 46	sseqtr4d 3605	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (◡𝐹 “ 𝑉) ⊆ ∪ 𝑥)
48	47	sselda 3568	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) → 𝑧 ∈ ∪ 𝑥)
49		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉))
50		ineq1 3769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑡 → (𝑦 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉)))
51	50	eqeq2d 2620	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑡 → ((𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)) ↔ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉))))
52	51	rspcev 3282	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ 𝑥 ∧ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉))) → ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)))
53	49, 52	mpan2 703	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝑥 → ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)))
54	53	ad2antrl 760	. . . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑡 ∈ V
56	55	inex1 4727	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ V
57	18	elrnmpt 5293	. . . . . . . . . . . . . . . 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 223	. . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) ∧ (𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡)) → 𝑧 ∈ 𝑡)
61		simplr 788	. . . . . . . . . . . . . . 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 3760	. . . . . . . . . . . . . 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 2677	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑡 ∩ (◡𝐹 “ 𝑉))))
64	63	rspcev 3282	. . . . . . . . . . . . . 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 691	. . . . . . . . . . . . 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 3014	. . . . . . . . . . . 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 4376	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑥 ↔ ∃𝑡 ∈ 𝑥 𝑧 ∈ 𝑡)
68		eluni2 4376	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ↔ ∃𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))𝑧 ∈ 𝑤)
69	66, 67, 68	3imtr4g 284	. . . . . . . . . . 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 449	. . . . . . . . 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 3574	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (◡𝐹 “ 𝑉) ⊆ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))))
73	41, 72	eqssd 3585	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (◡𝐹 “ 𝑉))
74		eldifsn 4260	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))}) ↔ (𝑧 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∧ 𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉))))
75		vex 3176	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
76	18	elrnmpt 5293	. . . . . . . . . . . . . . 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 1934	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑦 → (𝑦 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉)))
79	78	necon3ai 2807	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∩ (◡𝐹 “ 𝑉)) ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) → ¬ 𝑡 = 𝑦)
80		simpllr 795	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ (𝑆‘𝑈))
81		simplr 788	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑡 ∈ 𝑥)
82		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
83	7	cvmsdisj 30506	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (𝑆‘𝑈) ∧ 𝑡 ∈ 𝑥 ∧ 𝑦 ∈ 𝑥) → (𝑡 = 𝑦 ∨ (𝑡 ∩ 𝑦) = ∅))
84	80, 81, 82, 83	syl3anc 1318	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝑡 = 𝑦 ∨ (𝑡 ∩ 𝑦) = ∅))
85	84	ord 391	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (¬ 𝑡 = 𝑦 → (𝑡 ∩ 𝑦) = ∅))
86		inss1 3795	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) ⊆ (𝑡 ∩ 𝑦)
87		sseq0 3927	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) ⊆ (𝑡 ∩ 𝑦) ∧ (𝑡 ∩ 𝑦) = ∅) → ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅)
88	86, 87	mpan 702	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∩ 𝑦) = ∅ → ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅)
89	79, 85, 88	syl56 35	. . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → (𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) ↔ (𝑦 ∩ (◡𝐹 “ 𝑉)) ≠ (𝑡 ∩ (◡𝐹 “ 𝑉))))
91		ineq2 3770	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ (𝑦 ∩ (◡𝐹 “ 𝑉))))
92		inindir 3793	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ (𝑦 ∩ (◡𝐹 “ 𝑉)))
93	91, 92	syl6eqr 2662	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)))
94	93	eqeq1d 2612	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → (((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅ ↔ ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅))
95	90, 94	imbi12d 333	. . . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . . 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 3013	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 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 446	. . . . . . . . . . . 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 231	. . . . . . . . . . 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 2948	. . . . . . . . . 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 3795	. . . . . . . . . . . . 13 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ 𝑡
103		resabs1 5347	. . . . . . . . . . . . 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 30507	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝑆‘𝑈) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡) ∈ ((𝐶 ↾t 𝑡)Homeo(𝐽 ↾t 𝑈)))
106	105	adantll 746	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝐶 ∈ Top)
108	9	sselda 3568	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 ∈ 𝐶)
109		elssuni 4403	. . . . . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝐶 = ∪ 𝐶
112	111	restuni 20776	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ Top ∧ 𝑡 ⊆ ∪ 𝐶) → 𝑡 = ∪ (𝐶 ↾t 𝑡))
113	107, 110, 112	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 = ∪ (𝐶 ↾t 𝑡))
114	102, 113	syl5sseq 3616	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ ∪ (𝐶 ↾t 𝑡))
115		eqid 2610	. . . . . . . . . . . . . 14 ⊢ ∪ (𝐶 ↾t 𝑡) = ∪ (𝐶 ↾t 𝑡)
116	115	hmeores 21384	. . . . . . . . . . . . 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 691	. . . . . . . . . . . 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 2693	. . . . . . . . . . 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 476	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 ∈ 𝑥)
121		restabs 20779	. . . . . . . . . . . . 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 1318	. . . . . . . . . . . 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 3767	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) = ((◡𝐹 “ 𝑉) ∩ 𝑡)
124		cnvresima 5541	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝐹 ↾ 𝑡) “ 𝑉) = ((◡𝐹 “ 𝑉) ∩ 𝑡)
125	123, 124	eqtr4i 2635	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (◡(𝐹 ↾ 𝑡) “ 𝑉)
126	125	imaeq2i 5383	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉))) = ((𝐹 ↾ 𝑡) “ (◡(𝐹 ↾ 𝑡) “ 𝑉))
127	3	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝐹 ∈ (𝐶 CovMap 𝐽))
128		simplr 788	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑥 ∈ (𝑆‘𝑈))
129	7	cvmsf1o 30508	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥 ∈ (𝑆‘𝑈) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑈)
130	127, 128, 120, 129	syl3anc 1318	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑈)
131		f1ofo 6057	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑉 ⊆ 𝑈)
134		foimacnv 6067	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ↾ 𝑡):𝑡–onto→𝑈 ∧ 𝑉 ⊆ 𝑈) → ((𝐹 ↾ 𝑡) “ (◡(𝐹 ↾ 𝑡) “ 𝑉)) = 𝑉)
135	132, 133, 134	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐹 ↾ 𝑡) “ (◡(𝐹 ↾ 𝑡) “ 𝑉)) = 𝑉)
136	126, 135	syl5eq 2656	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉))) = 𝑉)
137	136	oveq2d 6565	. . . . . . . . . . . . 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 30497	. . . . . . . . . . . . . . . 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 30500	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽)
141	140	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑈 ∈ 𝐽)
142		restabs 20779	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑉 ⊆ 𝑈 ∧ 𝑈 ∈ 𝐽) → ((𝐽 ↾t 𝑈) ↾t 𝑉) = (𝐽 ↾t 𝑉))
143	139, 42, 141, 142	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ((𝐽 ↾t 𝑈) ↾t 𝑉) = (𝐽 ↾t 𝑉))
144	143	adantr 480	. . . . . . . . . . . . 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 2644	. . . . . . . . . . . 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 6567	. . . . . . . . . . 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 2690	. . . . . . . . . 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 553	. . . . . . . . 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 2949	. . . . . . . 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 2908	. . . . . . . . 9 ⊢ ∀𝑡 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ V
151	50	cbvmptv 4678	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (𝑡 ∈ 𝑥 ↦ (𝑡 ∩ (◡𝐹 “ 𝑉)))
152		sneq 4135	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → {𝑤} = {(𝑡 ∩ (◡𝐹 “ 𝑉))})
153	152	difeq2d 3690	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {𝑤}) = (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))}))
154		ineq1 3769	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝑤 ∩ 𝑧) = ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧))
155	154	eqeq1d 2612	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝑤 ∩ 𝑧) = ∅ ↔ ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅))
156	153, 155	raleqbidv 3129	. . . . . . . . . . 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 5312	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝐹 ↾ 𝑤) = (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))))
158		oveq2 6557	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝐶 ↾t 𝑤) = (𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉))))
159	158	oveq1d 6564	. . . . . . . . . . . 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 2682	. . . . . . . . . . 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 743	. . . . . . . . . 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 6276	. . . . . . . . 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 223	. . . . . . 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 553	. . . . . 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 30494	. . . . . . . 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 30502	. . . . . . 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 1238	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∈ (𝑆‘𝑉))
170		ne0i 3880	. . . . 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 449	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → (𝑥 ∈ (𝑆‘𝑈) → (𝑆‘𝑉) ≠ ∅))
173	172	exlimdv 1848	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → (∃𝑥 𝑥 ∈ (𝑆‘𝑈) → (𝑆‘𝑉) ≠ ∅))
174	1, 173	syl5bi 231	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → ((𝑆‘𝑈) ≠ ∅ → (𝑆‘𝑉) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↾_t crest 15904  Topctop 20517   Cn ccn 20838  Homeochmeo 21366   CovMap ccvm 30491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841  df-hmeo 21368  df-cvm 30492

This theorem is referenced by:  cvmcov2  30511