Theorem cvmsss2 35279

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 4353	. 2 ⊢ ((𝑆‘𝑈) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆‘𝑈))
2		simpl2 1193	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑉 ∈ 𝐽)
3		simpl1 1192	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
4		cvmtop1 35265	. . . . . . . . . . . 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 35272	. . . . . . . . . . . 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 3983	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐶)
11		cvmcn 35267	. . . . . . . . . . . . 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 23273	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝑉 ∈ 𝐽) → (◡𝐹 “ 𝑉) ∈ 𝐶)
14	12, 2, 13	syl2anc 584	. . . . . . . . . . 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 22905	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Top ∧ 𝑦 ∈ 𝐶 ∧ (◡𝐹 “ 𝑉) ∈ 𝐶) → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝐶)
17	6, 10, 15, 16	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝐶)
18	17	fmpttd 7135	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))):𝑥⟶𝐶)
19	18	frnd 6744	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝐶)
20	7	cvmsn0 35273	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑆‘𝑈) → 𝑥 ≠ ∅)
21	20	adantl 481	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑥 ≠ ∅)
22		dmmptg 6262	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ V → dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = 𝑥)
23		inex1g 5319	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑥 → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ V)
24	22, 23	mprg 3067	. . . . . . . . . . 11 ⊢ dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = 𝑥
25	24	eqeq1i 2742	. . . . . . . . . 10 ⊢ (dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅ ↔ 𝑥 = ∅)
26		dm0rn0 5935	. . . . . . . . . 10 ⊢ (dom (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅ ↔ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅)
27	25, 26	bitr3i 277	. . . . . . . . 9 ⊢ (𝑥 = ∅ ↔ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = ∅)
28	27	necon3bii 2993	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ≠ ∅)
29	21, 28	sylib 218	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ≠ ∅)
30	19, 29	jca 511	. . . . . 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 4238	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ (◡𝐹 “ 𝑉)) ⊆ (◡𝐹 “ 𝑉)
32		elpw2g 5333	. . . . . . . . . . . . 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 258	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∩ (◡𝐹 “ 𝑉)) ∈ 𝒫 (◡𝐹 “ 𝑉))
35	34	fmpttd 7135	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))):𝑥⟶𝒫 (◡𝐹 “ 𝑉))
36	35	frnd 6744	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝒫 (◡𝐹 “ 𝑉))
37		sspwuni 5100	. . . . . . . . 9 ⊢ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ 𝒫 (◡𝐹 “ 𝑉) ↔ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ (◡𝐹 “ 𝑉))
38	36, 37	sylib 218	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ⊆ (◡𝐹 “ 𝑉))
39		simpl3 1194	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑉 ⊆ 𝑈)
40		imass2 6120	. . . . . . . . . . . 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 35274	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑆‘𝑈) → ∪ 𝑥 = (◡𝐹 “ 𝑈))
43	42	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∪ 𝑥 = (◡𝐹 “ 𝑈))
44	41, 43	sseqtrrd 4021	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (◡𝐹 “ 𝑉) ⊆ ∪ 𝑥)
45	44	sselda 3983	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) → 𝑧 ∈ ∪ 𝑥)
46		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉))
47		ineq1 4213	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑡 → (𝑦 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉)))
48	47	rspceeqv 3645	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝑥 ∧ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉))) → ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)))
49	46, 48	mpan2 691	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑥 → ∃𝑦 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) = (𝑦 ∩ (◡𝐹 “ 𝑉)))
50	49	ad2antrl 728	. . . . . . . . . . . . 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 3484	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
52	51	inex1 5317	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ V
53		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))
54	53	elrnmpt 5969	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . 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 773	. . . . . . . . . . . . 13 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑧 ∈ (◡𝐹 “ 𝑉)) ∧ (𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡)) → 𝑧 ∈ 𝑡)
58		simplr 769	. . . . . . . . . . . . 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 4200	. . . . . . . . . . . 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 2830	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑡 ∩ (◡𝐹 “ 𝑉))))
61	60	rspcev 3622	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 3156	. . . . . . . . . 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 4911	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑥 ↔ ∃𝑡 ∈ 𝑥 𝑧 ∈ 𝑡)
65		eluni2 4911	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ↔ ∃𝑤 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉)))𝑧 ∈ 𝑤)
66	63, 64, 65	3imtr4g 296	. . . . . . . . 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 4005	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ∪ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (◡𝐹 “ 𝑉))
69		eldifsn 4786	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))}) ↔ (𝑧 ∈ ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∧ 𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉))))
70		vex 3484	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
71	53	elrnmpt 5969	. . . . . . . . . . . . . . 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 2019	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑦 → (𝑦 ∩ (◡𝐹 “ 𝑉)) = (𝑡 ∩ (◡𝐹 “ 𝑉)))
74	73	necon3ai 2965	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∩ (◡𝐹 “ 𝑉)) ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) → ¬ 𝑡 = 𝑦)
75		simpllr 776	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ (𝑆‘𝑈))
76		simplr 769	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑡 ∈ 𝑥)
77		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
78	7	cvmsdisj 35275	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (𝑆‘𝑈) ∧ 𝑡 ∈ 𝑥 ∧ 𝑦 ∈ 𝑥) → (𝑡 = 𝑦 ∨ (𝑡 ∩ 𝑦) = ∅))
79	75, 76, 77, 78	syl3anc 1373	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝑡 = 𝑦 ∨ (𝑡 ∩ 𝑦) = ∅))
80	79	ord 865	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (¬ 𝑡 = 𝑦 → (𝑡 ∩ 𝑦) = ∅))
81		inss1 4237	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) ⊆ (𝑡 ∩ 𝑦)
82		sseq0 4403	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) ⊆ (𝑡 ∩ 𝑦) ∧ (𝑡 ∩ 𝑦) = ∅) → ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅)
83	81, 82	mpan 690	. . . . . . . . . . . . . . . . 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 3003	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → (𝑧 ≠ (𝑡 ∩ (◡𝐹 “ 𝑉)) ↔ (𝑦 ∩ (◡𝐹 “ 𝑉)) ≠ (𝑡 ∩ (◡𝐹 “ 𝑉))))
86		ineq2 4214	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ (𝑦 ∩ (◡𝐹 “ 𝑉))))
87		inindir 4236	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ (𝑦 ∩ (◡𝐹 “ 𝑉)))
88	86, 87	eqtr4di 2795	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)))
89	88	eqeq1d 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑦 ∩ (◡𝐹 “ 𝑉)) → (((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅ ↔ ((𝑡 ∩ 𝑦) ∩ (◡𝐹 “ 𝑉)) = ∅))
90	85, 89	imbi12d 344	. . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . 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	biimtrid 242	. . . . . . . . . . . . 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 410	. . . . . . . . . . . 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	biimtrid 242	. . . . . . . . . . 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 3145	. . . . . . . . . 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 4237	. . . . . . . . . . . . 13 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ 𝑡
98		resabs1 6024	. . . . . . . . . . . . 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 35276	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝑆‘𝑈) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡) ∈ ((𝐶 ↾t 𝑡)Homeo(𝐽 ↾t 𝑈)))
101	100	adantll 714	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝐶 ∈ Top)
103	9	sselda 3983	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 ∈ 𝐶)
104		elssuni 4937	. . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝐶 = ∪ 𝐶
107	106	restuni 23170	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ Top ∧ 𝑡 ⊆ ∪ 𝐶) → 𝑡 = ∪ (𝐶 ↾t 𝑡))
108	102, 105, 107	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 = ∪ (𝐶 ↾t 𝑡))
109	97, 108	sseqtrid 4026	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝑡 ∩ (◡𝐹 “ 𝑉)) ⊆ ∪ (𝐶 ↾t 𝑡))
110		eqid 2737	. . . . . . . . . . . . . 14 ⊢ ∪ (𝐶 ↾t 𝑡) = ∪ (𝐶 ↾t 𝑡)
111	110	hmeores 23779	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 2846	. . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑡 ∈ 𝑥)
116		restabs 23173	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 4209	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) = ((◡𝐹 “ 𝑉) ∩ 𝑡)
119		cnvresima 6250	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝐹 ↾ 𝑡) “ 𝑉) = ((◡𝐹 “ 𝑉) ∩ 𝑡)
120	118, 119	eqtr4i 2768	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∩ (◡𝐹 “ 𝑉)) = (◡(𝐹 ↾ 𝑡) “ 𝑉)
121	120	imaeq2i 6076	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉))) = ((𝐹 ↾ 𝑡) “ (◡(𝐹 ↾ 𝑡) “ 𝑉))
122	3	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝐹 ∈ (𝐶 CovMap 𝐽))
123		simplr 769	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑥 ∈ (𝑆‘𝑈))
124	7	cvmsf1o 35277	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥 ∈ (𝑆‘𝑈) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑈)
125	122, 123, 115, 124	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑈)
126		f1ofo 6855	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → 𝑉 ⊆ 𝑈)
129		foimacnv 6865	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ↾ 𝑡):𝑡–onto→𝑈 ∧ 𝑉 ⊆ 𝑈) → ((𝐹 ↾ 𝑡) “ (◡(𝐹 ↾ 𝑡) “ 𝑉)) = 𝑉)
130	127, 128, 129	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐹 ↾ 𝑡) “ (◡(𝐹 ↾ 𝑡) “ 𝑉)) = 𝑉)
131	121, 130	eqtrid 2789	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) ∧ 𝑡 ∈ 𝑥) → ((𝐹 ↾ 𝑡) “ (𝑡 ∩ (◡𝐹 “ 𝑉))) = 𝑉)
132	131	oveq2d 7447	. . . . . . . . . . . . 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 35266	. . . . . . . . . . . . . . . 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 35269	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽)
136	135	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → 𝑈 ∈ 𝐽)
137		restabs 23173	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑉 ⊆ 𝑈 ∧ 𝑈 ∈ 𝐽) → ((𝐽 ↾t 𝑈) ↾t 𝑉) = (𝐽 ↾t 𝑉))
138	134, 39, 136, 137	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ((𝐽 ↾t 𝑈) ↾t 𝑉) = (𝐽 ↾t 𝑉))
139	138	adantr 480	. . . . . . . . . . . . 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 2777	. . . . . . . . . . . 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 7449	. . . . . . . . . . 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 2843	. . . . . . . . . 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 511	. . . . . . . . 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 3146	. . . . . . . 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 3065	. . . . . . . . 9 ⊢ ∀𝑡 ∈ 𝑥 (𝑡 ∩ (◡𝐹 “ 𝑉)) ∈ V
146	47	cbvmptv 5255	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) = (𝑡 ∈ 𝑥 ↦ (𝑡 ∩ (◡𝐹 “ 𝑉)))
147		sneq 4636	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → {𝑤} = {(𝑡 ∩ (◡𝐹 “ 𝑉))})
148	147	difeq2d 4126	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {𝑤}) = (ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∖ {(𝑡 ∩ (◡𝐹 “ 𝑉))}))
149		ineq1 4213	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝑤 ∩ 𝑧) = ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧))
150	149	eqeq1d 2739	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → ((𝑤 ∩ 𝑧) = ∅ ↔ ((𝑡 ∩ (◡𝐹 “ 𝑉)) ∩ 𝑧) = ∅))
151	148, 150	raleqbidv 3346	. . . . . . . . . . 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 5992	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝐹 ↾ 𝑤) = (𝐹 ↾ (𝑡 ∩ (◡𝐹 “ 𝑉))))
153		oveq2 7439	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑡 ∩ (◡𝐹 “ 𝑉)) → (𝐶 ↾t 𝑤) = (𝐶 ↾t (𝑡 ∩ (◡𝐹 “ 𝑉))))
154	153	oveq1d 7446	. . . . . . . . . . . 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 2835	. . . . . . . . . . 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 632	. . . . . . . . . 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 7114	. . . . . . . . 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 234	. . . . . . 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 511	. . . . . 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 35263	. . . . . . . 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 35271	. . . . . . 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 1343	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ (◡𝐹 “ 𝑉))) ∈ (𝑆‘𝑉))
165	164	ne0d 4342	. . . 4 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑥 ∈ (𝑆‘𝑈)) → (𝑆‘𝑉) ≠ ∅)
166	165	ex 412	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → (𝑥 ∈ (𝑆‘𝑈) → (𝑆‘𝑉) ≠ ∅))
167	166	exlimdv 1933	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → (∃𝑥 𝑥 ∈ (𝑆‘𝑈) → (𝑆‘𝑉) ≠ ∅))
168	1, 167	biimtrid 242	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → ((𝑆‘𝑈) ≠ ∅ → (𝑆‘𝑉) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  {csn 4626  ∪ cuni 4907   ↦ cmpt 5225  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   ↾ cres 5687   “ cima 5688  –onto→wfo 6559  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431   ↾_t crest 17465  Topctop 22899   Cn ccn 23232  Homeochmeo 23761   CovMap ccvm 35260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-map 8868  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cn 23235  df-hmeo 23763  df-cvm 35261

This theorem is referenced by:  cvmcov2  35280