Theorem cvmsss2 32634

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 4260	. 2 ⊢ ((𝑆‘𝑈) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆‘𝑈))
2		simpl2 1189	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑉 ∈ 𝐽)
3		simpl1 1188	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
4		cvmtop1 32620	. . . . . . . . . . . 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 484	. . . . . . . . . 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 32627	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑆‘𝑈) → 𝑥 ⊆ 𝐶)
9	8	adantl 485	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑥 ⊆ 𝐶)
10	9	sselda 3915	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐶)
11		cvmcn 32622	. . . . . . . . . . . . 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 21870	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝑉 ∈ 𝐽) → (◡𝐹 “ 𝑉) ∈ 𝐶)
14	12, 2, 13	syl2anc 587	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (◡𝐹 “ 𝑉) ∈ 𝐶)
15	14	adantr 484	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → (◡𝐹 “ 𝑉) ∈ 𝐶)
16		inopn 21504	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Top ∧ 𝑦 ∈ 𝐶 ∧ (◡𝐹 “ 𝑉) ∈ 𝐶) → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝐶)
17	6, 10, 15, 16	syl3anc 1368	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝐶)
18	17	fmpttd 6856	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))):𝑥⟶𝐶)
19	18	frnd 6494	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝐶)
20	7	cvmsn0 32628	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑆‘𝑈) → 𝑥 ≠ ∅)
21	20	adantl 485	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑥 ≠ ∅)
22		dmmptg 6063	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ V → dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = 𝑥)
23		inex1g 5187	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑥 → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ V)
24	22, 23	mprg 3120	. . . . . . . . . . 11 ⊢ dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = 𝑥
25	24	eqeq1i 2803	. . . . . . . . . 10 ⊢ (dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅ ↔ 𝑥 = ∅)
26		dm0rn0 5759	. . . . . . . . . 10 ⊢ (dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅ ↔ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅)
27	25, 26	bitr3i 280	. . . . . . . . 9 ⊢ (𝑥 = ∅ ↔ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅)
28	27	necon3bii 3039	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ≠ ∅)
29	21, 28	sylib 221	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ≠ ∅)
30	19, 29	jca 515	. . . . . 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))) ≠ ∅))
31		inss2 4156	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ (◡𝐹 “ 𝑉)) ⊆ (◡𝐹 “ 𝑉)
32		elpw2g 5211	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ 𝑉) ∈ 𝐶 → ((𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝒫 (◡𝐹 “ 𝑉) ↔ (𝑦 ∩ (◡𝐹 “ 𝑉)) ⊆ (◡𝐹 “ 𝑉)))
33	15, 32	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)))
34	31, 33	mpbiri 261	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝒫 (◡𝐹 “ 𝑉))
35	34	fmpttd 6856	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))):𝑥⟶𝒫 (◡𝐹 “ 𝑉))
36	35	frnd 6494	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝒫 (◡𝐹 “ 𝑉))
37		sspwuni 4985	. . . . . . . . 9 ⊢ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝒫 (◡𝐹 “ 𝑉) ↔ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ (◡𝐹 “ 𝑉))
38	36, 37	sylib 221	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ (◡𝐹 “ 𝑉))
39		simpl3 1190	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑉 ⊆ 𝑈)
40		imass2 5932	. . . . . . . . . . . 12 ⊢ (𝑉 ⊆ 𝑈 → (◡𝐹 “ 𝑉) ⊆ (◡𝐹 “ 𝑈))
41	39, 40	syl 17	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (◡𝐹 “ 𝑉) ⊆ (◡𝐹 “ 𝑈))
42	7	cvmsuni 32629	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑆‘𝑈) → ∪ 𝑥 = (◡𝐹 “ 𝑈))
43	42	adantl 485	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∪ 𝑥 = (◡𝐹 “ 𝑈))
44	41, 43	sseqtrrd 3956	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (◡𝐹 “ 𝑉) ⊆ ∪ 𝑥)
45	44	sselda 3915	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) → 𝑧 ∈ ∪ 𝑥)
46		eqid 2798	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉))
47		ineq1 4131	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑡 → (𝑦 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉)))
48	47	rspceeqv 3586	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝑥 ∧ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉))) → ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)))
49	46, 48	mpan2 690	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑥 → ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)))
50	49	ad2antrl 727	. . . . . . . . . . . . 13 ⊢ (((((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)))
51		vex 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
52	51	inex1 5185	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ V
53		eqid 2798	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))
54	53	elrnmpt 5792	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ V → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ↔ ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉))))
55	52, 54	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ↔ ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)))
56	50, 55	sylibr 237	. . . . . . . . . . . 12 ⊢ (((((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))))
57		simprr 772	. . . . . . . . . . . . 13 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) ∧ (𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡)) → 𝑧 ∈ 𝑡)
58		simplr 768	. . . . . . . . . . . . 13 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) ∧ (𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡)) → 𝑧 ∈ (◡𝐹 “ 𝑉))
59	57, 58	elind 4121	. . . . . . . . . . . 12 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) ∧ (𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡)) → 𝑧 ∈ (𝑡 ∩ (◡𝐹 “ 𝑉)))
60		eleq2 2878	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑡 ∩ (◡𝐹 “ 𝑉))))
61	60	rspcev 3571	. . . . . . . . . . . 12 ⊢ (((𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∧ 𝑧 ∈ (𝑡 ∩ (◡𝐹 “ 𝑉))) → ∃𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))𝑧 ∈ 𝑤)
62	56, 59, 61	syl2anc 587	. . . . . . . . . . 11 ⊢ (((((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)
63	62	rexlimdvaa 3244	. . . . . . . . . 10 ⊢ ((((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))
64		eluni2 4804	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑥 ↔ ∃𝑡 ∈ 𝑥 𝑧 ∈ 𝑡)
65		eluni2 4804	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ↔ ∃𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))𝑧 ∈ 𝑤)
66	63, 64, 65	3imtr4g 299	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) → (𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))))
67	45, 66	mpd 15	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) → 𝑧 ∈ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))))
68	38, 67	eqelssd 3936	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (◡𝐹 “ 𝑉))
69		eldifsn 4680	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))}) ↔ (𝑧 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∧ 𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉))))
70		vex 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
71	53	elrnmpt 5792	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ↔ ∃𝑦 ∈ 𝑥 𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉))))
72	70, 71	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ↔ ∃𝑦 ∈ 𝑥 𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)))
73	47	equcoms 2027	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑦 → (𝑦 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉)))
74	73	necon3ai 3012	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∩ (◡𝐹 “ 𝑉)) ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) → ¬ 𝑡 = 𝑦)
75		simpllr 775	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ (𝑆‘𝑈))
76		simplr 768	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑡 ∈ 𝑥)
77		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
78	7	cvmsdisj 32630	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (𝑆‘𝑈) ∧ 𝑡 ∈ 𝑥 ∧ 𝑦 ∈ 𝑥) → (𝑡 = 𝑦 ∨ (𝑡 ∩ 𝑦) = ∅))
79	75, 76, 77, 78	syl3anc 1368	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝑡 = 𝑦 ∨ (𝑡 ∩ 𝑦) = ∅))
80	79	ord 861	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (¬ 𝑡 = 𝑦 → (𝑡 ∩ 𝑦) = ∅))
81		inss1 4155	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) ⊆ (𝑡 ∩ 𝑦)
82		sseq0 4307	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) ⊆ (𝑡 ∩ 𝑦) ∧ (𝑡 ∩ 𝑦) = ∅) → ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅)
83	81, 82	mpan 689	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∩ 𝑦) = ∅ → ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅)
84	74, 80, 83	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)) = ∅))
85		neeq1 3049	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → (𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) ↔ (𝑦 ∩ (◡𝐹 “ 𝑉)) ≠ (𝑡 ∩ (◡𝐹 “ 𝑉))))
86		ineq2 4133	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ (𝑦 ∩ (◡𝐹 “ 𝑉))))
87		inindir 4154	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ (𝑦 ∩ (◡𝐹 “ 𝑉)))
88	86, 87	eqtr4di 2851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)))
89	88	eqeq1d 2800	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → (((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅ ↔ ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅))
90	85, 89	imbi12d 348	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → ((𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅) ↔ ((𝑦 ∩ (◡𝐹 “ 𝑉)) ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅)))
91	84, 90	syl5ibrcom 250	. . . . . . . . . . . . . . 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) = ∅)))
92	91	rexlimdva 3243	. . . . . . . . . . . . . 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) = ∅)))
93	72, 92	syl5bi 245	. . . . . . . . . . . . 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) = ∅)))
94	93	impd 414	. . . . . . . . . . . 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) = ∅))
95	69, 94	syl5bi 245	. . . . . . . . . . 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) = ∅))
96	95	ralrimiv 3148	. . . . . . . . . 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) = ∅)
97		inss1 4155	. . . . . . . . . . . . 13 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ 𝑡
98		resabs1 5848	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ 𝑡 → ((𝐹 ↾ 𝑡) ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) = (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))))
99	97, 98	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ 𝑡) ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))) = (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉)))
100	7	cvmshmeo 32631	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝑆‘𝑈) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡) ∈ ((𝐶 ↾t 𝑡)Homeo(𝐽 ↾t 𝑈)))
101	100	adantll 713	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡) ∈ ((𝐶 ↾t 𝑡)Homeo(𝐽 ↾t 𝑈)))
102	5	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝐶 ∈ Top)
103	9	sselda 3915	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 ∈ 𝐶)
104		elssuni 4830	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝐶 → 𝑡 ⊆ ∪ 𝐶)
105	103, 104	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 ⊆ ∪ 𝐶)
106		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝐶 = ∪ 𝐶
107	106	restuni 21767	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ Top ∧ 𝑡 ⊆ ∪ 𝐶) → 𝑡 = ∪ (𝐶 ↾t 𝑡))
108	102, 105, 107	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 = ∪ (𝐶 ↾t 𝑡))
109	97, 108	sseqtrid 3967	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ ∪ (𝐶 ↾t 𝑡))
110		eqid 2798	. . . . . . . . . . . . . 14 ⊢ ∪ (𝐶 ↾t 𝑡) = ∪ (𝐶 ↾t 𝑡)
111	110	hmeores 22376	. . . . . . . . . . . . 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))))))
112	101, 109, 111	syl2anc 587	. . . . . . . . . . . 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))))))
113	99, 112	eqeltrrid 2895	. . . . . . . . . . 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))))))
114	97	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ 𝑡)
115		simpr 488	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 ∈ 𝑥)
116		restabs 21770	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ Top ∧ (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ 𝑡 ∧ 𝑡 ∈ 𝑥) → ((𝐶 ↾t 𝑡) ↾t (𝑡 ∩ (◡𝐹 “ 𝑉))) = (𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉))))
117	102, 114, 115, 116	syl3anc 1368	. . . . . . . . . . . 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))))
118		incom 4128	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) = ((◡𝐹 “ 𝑉) ∩ 𝑡)
119		cnvresima 6054	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝐹 ↾ 𝑡) “ 𝑉) = ((◡𝐹 “ 𝑉) ∩ 𝑡)
120	118, 119	eqtr4i 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (◡(𝐹 ↾ 𝑡) “ 𝑉)
121	120	imaeq2i 5894	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉))) = ((𝐹 ↾ 𝑡) “ (◡(𝐹 ↾ 𝑡) “ 𝑉))
122	3	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝐹 ∈ (𝐶 CovMap 𝐽))
123		simplr 768	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑥 ∈ (𝑆‘𝑈))
124	7	cvmsf1o 32632	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥 ∈ (𝑆‘𝑈) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑈)
125	122, 123, 115, 124	syl3anc 1368	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑈)
126		f1ofo 6597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑈 → (𝐹 ↾ 𝑡):𝑡–onto→𝑈)
127	125, 126	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡):𝑡–onto→𝑈)
128	39	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑉 ⊆ 𝑈)
129		foimacnv 6607	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ↾ 𝑡):𝑡–onto→𝑈 ∧ 𝑉 ⊆ 𝑈) → ((𝐹 ↾ 𝑡) “ (◡(𝐹 ↾ 𝑡) “ 𝑉)) = 𝑉)
130	127, 128, 129	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐹 ↾ 𝑡) “ (◡(𝐹 ↾ 𝑡) “ 𝑉)) = 𝑉)
131	121, 130	syl5eq 2845	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉))) = 𝑉)
132	131	oveq2d 7151	. . . . . . . . . . . . 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))
133		cvmtop2 32621	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
134	3, 133	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝐽 ∈ Top)
135	7	cvmsrcl 32624	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽)
136	135	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑈 ∈ 𝐽)
137		restabs 21770	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑉 ⊆ 𝑈 ∧ 𝑈 ∈ 𝐽) → ((𝐽 ↾t 𝑈) ↾t 𝑉) = (𝐽 ↾t 𝑉))
138	134, 39, 136, 137	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ((𝐽 ↾t 𝑈) ↾t 𝑉) = (𝐽 ↾t 𝑉))
139	138	adantr 484	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐽 ↾t 𝑈) ↾t 𝑉) = (𝐽 ↾t 𝑉))
140	132, 139	eqtrd 2833	. . . . . . . . . . . 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))
141	117, 140	oveq12d 7153	. . . . . . . . . . 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)))
142	113, 141	eleqtrd 2892	. . . . . . . . . 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)))
143	96, 142	jca 515	. . . . . . . . 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))))
144	143	ralrimiva 3149	. . . . . . . 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))))
145	52	rgenw 3118	. . . . . . . . 9 ⊢ ∀𝑡 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ V
146	47	cbvmptv 5133	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (𝑡 ∈ 𝑥 ↦ (𝑡 ∩ (◡𝐹 “ 𝑉)))
147		sneq 4535	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → {𝑤} = {(𝑡 ∩ (◡𝐹 “ 𝑉))})
148	147	difeq2d 4050	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {𝑤}) = (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))}))
149		ineq1 4131	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝑤 ∩ 𝑧) = ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧))
150	149	eqeq1d 2800	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝑤 ∩ 𝑧) = ∅ ↔ ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅))
151	148, 150	raleqbidv 3354	. . . . . . . . . . 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) = ∅))
152		reseq2 5813	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝐹 ↾ 𝑤) = (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))))
153		oveq2 7143	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝐶 ↾t 𝑤) = (𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉))))
154	153	oveq1d 7150	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝐶 ↾t 𝑤)Homeo(𝐽 ↾t 𝑉)) = ((𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉)))Homeo(𝐽 ↾t 𝑉)))
155	152, 154	eleq12d 2884	. . . . . . . . . . 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))))
156	151, 155	anbi12d 633	. . . . . . . . . 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)))))
157	146, 156	ralrnmptw 6837	. . . . . . . . 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)))))
158	145, 157	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))))
159	144, 158	sylibr 237	. . . . . . 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))))
160	68, 159	jca 515	. . . . . 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)))))
161	7	cvmscbv 32618	. . . . . . . 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))))})
162	161	cvmsval 32626	. . . . . . 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)))))))
163	5, 162	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)))))))
164	2, 30, 160, 163	mpbir3and 1339	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∈ (𝑆‘𝑉))
165	164	ne0d 4251	. . . 4 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (𝑆‘𝑉) ≠ ∅)
166	165	ex 416	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → (𝑥 ∈ (𝑆‘𝑈) → (𝑆‘𝑉) ≠ ∅))
167	166	exlimdv 1934	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → (∃𝑥 𝑥 ∈ (𝑆‘𝑈) → (𝑆‘𝑉) ≠ ∅))
168	1, 167	syl5bi 245	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → ((𝑆‘𝑈) ≠ ∅ → (𝑆‘𝑉) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {csn 4525  ∪ cuni 4800   ↦ cmpt 5110  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   ↾ cres 5521   “ cima 5522  –onto→wfo 6322  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135   ↾_t crest 16686  Topctop 21498   Cn ccn 21829  Homeochmeo 22358   CovMap ccvm 32615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-fin 8496  df-fi 8859  df-rest 16688  df-topgen 16709  df-top 21499  df-topon 21516  df-bases 21551  df-cn 21832  df-hmeo 22360  df-cvm 32616

This theorem is referenced by:  cvmcov2  32635