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Theorem cvmsss2 35015
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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
Assertion
Ref Expression
cvmsss2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) → ((𝑆𝑈) ≠ ∅ → (𝑆𝑉) ≠ ∅))
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.
StepHypRef Expression
1 n0 4346 . 2 ((𝑆𝑈) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆𝑈))
2 simpl2 1189 . . . . . 6 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝑉𝐽)
3 simpl1 1188 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
4 cvmtop1 35001 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
53, 4syl 17 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝐶 ∈ Top)
65adantr 479 . . . . . . . . . 10 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑦𝑥) → 𝐶 ∈ Top)
7 cvmcov.1 . . . . . . . . . . . . 13 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
87cvmsss 35008 . . . . . . . . . . . 12 (𝑥 ∈ (𝑆𝑈) → 𝑥𝐶)
98adantl 480 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝑥𝐶)
109sselda 3976 . . . . . . . . . 10 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑦𝑥) → 𝑦𝐶)
11 cvmcn 35003 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
123, 11syl 17 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝐹 ∈ (𝐶 Cn 𝐽))
13 cnima 23213 . . . . . . . . . . . 12 ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝑉𝐽) → (𝐹𝑉) ∈ 𝐶)
1412, 2, 13syl2anc 582 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (𝐹𝑉) ∈ 𝐶)
1514adantr 479 . . . . . . . . . 10 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑦𝑥) → (𝐹𝑉) ∈ 𝐶)
16 inopn 22845 . . . . . . . . . 10 ((𝐶 ∈ Top ∧ 𝑦𝐶 ∧ (𝐹𝑉) ∈ 𝐶) → (𝑦 ∩ (𝐹𝑉)) ∈ 𝐶)
176, 10, 15, 16syl3anc 1368 . . . . . . . . 9 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑦𝑥) → (𝑦 ∩ (𝐹𝑉)) ∈ 𝐶)
1817fmpttd 7124 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))):𝑥𝐶)
1918frnd 6731 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ 𝐶)
207cvmsn0 35009 . . . . . . . . 9 (𝑥 ∈ (𝑆𝑈) → 𝑥 ≠ ∅)
2120adantl 480 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝑥 ≠ ∅)
22 dmmptg 6248 . . . . . . . . . . . 12 (∀𝑦𝑥 (𝑦 ∩ (𝐹𝑉)) ∈ V → dom (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = 𝑥)
23 inex1g 5320 . . . . . . . . . . . 12 (𝑦𝑥 → (𝑦 ∩ (𝐹𝑉)) ∈ V)
2422, 23mprg 3056 . . . . . . . . . . 11 dom (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = 𝑥
2524eqeq1i 2730 . . . . . . . . . 10 (dom (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = ∅ ↔ 𝑥 = ∅)
26 dm0rn0 5927 . . . . . . . . . 10 (dom (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = ∅ ↔ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = ∅)
2725, 26bitr3i 276 . . . . . . . . 9 (𝑥 = ∅ ↔ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = ∅)
2827necon3bii 2982 . . . . . . . 8 (𝑥 ≠ ∅ ↔ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ≠ ∅)
2921, 28sylib 217 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ≠ ∅)
3019, 29jca 510 . . . . . 6 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ 𝐶 ∧ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ≠ ∅))
31 inss2 4228 . . . . . . . . . . . 12 (𝑦 ∩ (𝐹𝑉)) ⊆ (𝐹𝑉)
32 elpw2g 5347 . . . . . . . . . . . . 13 ((𝐹𝑉) ∈ 𝐶 → ((𝑦 ∩ (𝐹𝑉)) ∈ 𝒫 (𝐹𝑉) ↔ (𝑦 ∩ (𝐹𝑉)) ⊆ (𝐹𝑉)))
3315, 32syl 17 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑦𝑥) → ((𝑦 ∩ (𝐹𝑉)) ∈ 𝒫 (𝐹𝑉) ↔ (𝑦 ∩ (𝐹𝑉)) ⊆ (𝐹𝑉)))
3431, 33mpbiri 257 . . . . . . . . . . 11 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑦𝑥) → (𝑦 ∩ (𝐹𝑉)) ∈ 𝒫 (𝐹𝑉))
3534fmpttd 7124 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))):𝑥⟶𝒫 (𝐹𝑉))
3635frnd 6731 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ 𝒫 (𝐹𝑉))
37 sspwuni 5104 . . . . . . . . 9 (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ 𝒫 (𝐹𝑉) ↔ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ (𝐹𝑉))
3836, 37sylib 217 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ (𝐹𝑉))
39 simpl3 1190 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝑉𝑈)
40 imass2 6107 . . . . . . . . . . . 12 (𝑉𝑈 → (𝐹𝑉) ⊆ (𝐹𝑈))
4139, 40syl 17 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (𝐹𝑉) ⊆ (𝐹𝑈))
427cvmsuni 35010 . . . . . . . . . . . 12 (𝑥 ∈ (𝑆𝑈) → 𝑥 = (𝐹𝑈))
4342adantl 480 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝑥 = (𝐹𝑈))
4441, 43sseqtrrd 4018 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (𝐹𝑉) ⊆ 𝑥)
4544sselda 3976 . . . . . . . . 9 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) → 𝑧 𝑥)
46 eqid 2725 . . . . . . . . . . . . . . 15 (𝑡 ∩ (𝐹𝑉)) = (𝑡 ∩ (𝐹𝑉))
47 ineq1 4203 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑡 → (𝑦 ∩ (𝐹𝑉)) = (𝑡 ∩ (𝐹𝑉)))
4847rspceeqv 3628 . . . . . . . . . . . . . . 15 ((𝑡𝑥 ∧ (𝑡 ∩ (𝐹𝑉)) = (𝑡 ∩ (𝐹𝑉))) → ∃𝑦𝑥 (𝑡 ∩ (𝐹𝑉)) = (𝑦 ∩ (𝐹𝑉)))
4946, 48mpan2 689 . . . . . . . . . . . . . 14 (𝑡𝑥 → ∃𝑦𝑥 (𝑡 ∩ (𝐹𝑉)) = (𝑦 ∩ (𝐹𝑉)))
5049ad2antrl 726 . . . . . . . . . . . . 13 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) ∧ (𝑡𝑥𝑧𝑡)) → ∃𝑦𝑥 (𝑡 ∩ (𝐹𝑉)) = (𝑦 ∩ (𝐹𝑉)))
51 vex 3465 . . . . . . . . . . . . . . 15 𝑡 ∈ V
5251inex1 5318 . . . . . . . . . . . . . 14 (𝑡 ∩ (𝐹𝑉)) ∈ V
53 eqid 2725 . . . . . . . . . . . . . . 15 (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))
5453elrnmpt 5958 . . . . . . . . . . . . . 14 ((𝑡 ∩ (𝐹𝑉)) ∈ V → ((𝑡 ∩ (𝐹𝑉)) ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ↔ ∃𝑦𝑥 (𝑡 ∩ (𝐹𝑉)) = (𝑦 ∩ (𝐹𝑉))))
5552, 54ax-mp 5 . . . . . . . . . . . . 13 ((𝑡 ∩ (𝐹𝑉)) ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ↔ ∃𝑦𝑥 (𝑡 ∩ (𝐹𝑉)) = (𝑦 ∩ (𝐹𝑉)))
5650, 55sylibr 233 . . . . . . . . . . . 12 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) ∧ (𝑡𝑥𝑧𝑡)) → (𝑡 ∩ (𝐹𝑉)) ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))))
57 simprr 771 . . . . . . . . . . . . 13 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) ∧ (𝑡𝑥𝑧𝑡)) → 𝑧𝑡)
58 simplr 767 . . . . . . . . . . . . 13 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) ∧ (𝑡𝑥𝑧𝑡)) → 𝑧 ∈ (𝐹𝑉))
5957, 58elind 4192 . . . . . . . . . . . 12 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) ∧ (𝑡𝑥𝑧𝑡)) → 𝑧 ∈ (𝑡 ∩ (𝐹𝑉)))
60 eleq2 2814 . . . . . . . . . . . . 13 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → (𝑧𝑤𝑧 ∈ (𝑡 ∩ (𝐹𝑉))))
6160rspcev 3606 . . . . . . . . . . . 12 (((𝑡 ∩ (𝐹𝑉)) ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∧ 𝑧 ∈ (𝑡 ∩ (𝐹𝑉))) → ∃𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))𝑧𝑤)
6256, 59, 61syl2anc 582 . . . . . . . . . . 11 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) ∧ (𝑡𝑥𝑧𝑡)) → ∃𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))𝑧𝑤)
6362rexlimdvaa 3145 . . . . . . . . . 10 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) → (∃𝑡𝑥 𝑧𝑡 → ∃𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))𝑧𝑤))
64 eluni2 4913 . . . . . . . . . 10 (𝑧 𝑥 ↔ ∃𝑡𝑥 𝑧𝑡)
65 eluni2 4913 . . . . . . . . . 10 (𝑧 ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ↔ ∃𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))𝑧𝑤)
6663, 64, 653imtr4g 295 . . . . . . . . 9 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) → (𝑧 𝑥𝑧 ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))))
6745, 66mpd 15 . . . . . . . 8 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) → 𝑧 ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))))
6838, 67eqelssd 3998 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = (𝐹𝑉))
69 eldifsn 4792 . . . . . . . . . . . 12 (𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))}) ↔ (𝑧 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∧ 𝑧 ≠ (𝑡 ∩ (𝐹𝑉))))
70 vex 3465 . . . . . . . . . . . . . . 15 𝑧 ∈ V
7153elrnmpt 5958 . . . . . . . . . . . . . . 15 (𝑧 ∈ V → (𝑧 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ↔ ∃𝑦𝑥 𝑧 = (𝑦 ∩ (𝐹𝑉))))
7270, 71ax-mp 5 . . . . . . . . . . . . . 14 (𝑧 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ↔ ∃𝑦𝑥 𝑧 = (𝑦 ∩ (𝐹𝑉)))
7347equcoms 2015 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑦 → (𝑦 ∩ (𝐹𝑉)) = (𝑡 ∩ (𝐹𝑉)))
7473necon3ai 2954 . . . . . . . . . . . . . . . . 17 ((𝑦 ∩ (𝐹𝑉)) ≠ (𝑡 ∩ (𝐹𝑉)) → ¬ 𝑡 = 𝑦)
75 simpllr 774 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → 𝑥 ∈ (𝑆𝑈))
76 simplr 767 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → 𝑡𝑥)
77 simpr 483 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → 𝑦𝑥)
787cvmsdisj 35011 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ (𝑆𝑈) ∧ 𝑡𝑥𝑦𝑥) → (𝑡 = 𝑦 ∨ (𝑡𝑦) = ∅))
7975, 76, 77, 78syl3anc 1368 . . . . . . . . . . . . . . . . . 18 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → (𝑡 = 𝑦 ∨ (𝑡𝑦) = ∅))
8079ord 862 . . . . . . . . . . . . . . . . 17 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → (¬ 𝑡 = 𝑦 → (𝑡𝑦) = ∅))
81 inss1 4227 . . . . . . . . . . . . . . . . . 18 ((𝑡𝑦) ∩ (𝐹𝑉)) ⊆ (𝑡𝑦)
82 sseq0 4401 . . . . . . . . . . . . . . . . . 18 ((((𝑡𝑦) ∩ (𝐹𝑉)) ⊆ (𝑡𝑦) ∧ (𝑡𝑦) = ∅) → ((𝑡𝑦) ∩ (𝐹𝑉)) = ∅)
8381, 82mpan 688 . . . . . . . . . . . . . . . . 17 ((𝑡𝑦) = ∅ → ((𝑡𝑦) ∩ (𝐹𝑉)) = ∅)
8474, 80, 83syl56 36 . . . . . . . . . . . . . . . 16 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → ((𝑦 ∩ (𝐹𝑉)) ≠ (𝑡 ∩ (𝐹𝑉)) → ((𝑡𝑦) ∩ (𝐹𝑉)) = ∅))
85 neeq1 2992 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑦 ∩ (𝐹𝑉)) → (𝑧 ≠ (𝑡 ∩ (𝐹𝑉)) ↔ (𝑦 ∩ (𝐹𝑉)) ≠ (𝑡 ∩ (𝐹𝑉))))
86 ineq2 4204 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑦 ∩ (𝐹𝑉)) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ((𝑡 ∩ (𝐹𝑉)) ∩ (𝑦 ∩ (𝐹𝑉))))
87 inindir 4226 . . . . . . . . . . . . . . . . . . 19 ((𝑡𝑦) ∩ (𝐹𝑉)) = ((𝑡 ∩ (𝐹𝑉)) ∩ (𝑦 ∩ (𝐹𝑉)))
8886, 87eqtr4di 2783 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑦 ∩ (𝐹𝑉)) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ((𝑡𝑦) ∩ (𝐹𝑉)))
8988eqeq1d 2727 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑦 ∩ (𝐹𝑉)) → (((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅ ↔ ((𝑡𝑦) ∩ (𝐹𝑉)) = ∅))
9085, 89imbi12d 343 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑦 ∩ (𝐹𝑉)) → ((𝑧 ≠ (𝑡 ∩ (𝐹𝑉)) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅) ↔ ((𝑦 ∩ (𝐹𝑉)) ≠ (𝑡 ∩ (𝐹𝑉)) → ((𝑡𝑦) ∩ (𝐹𝑉)) = ∅)))
9184, 90syl5ibrcom 246 . . . . . . . . . . . . . . 15 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → (𝑧 = (𝑦 ∩ (𝐹𝑉)) → (𝑧 ≠ (𝑡 ∩ (𝐹𝑉)) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅)))
9291rexlimdva 3144 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (∃𝑦𝑥 𝑧 = (𝑦 ∩ (𝐹𝑉)) → (𝑧 ≠ (𝑡 ∩ (𝐹𝑉)) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅)))
9372, 92biimtrid 241 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝑧 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) → (𝑧 ≠ (𝑡 ∩ (𝐹𝑉)) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅)))
9493impd 409 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝑧 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∧ 𝑧 ≠ (𝑡 ∩ (𝐹𝑉))) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅))
9569, 94biimtrid 241 . . . . . . . . . . 11 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))}) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅))
9695ralrimiv 3134 . . . . . . . . . 10 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))})((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅)
97 inss1 4227 . . . . . . . . . . . . 13 (𝑡 ∩ (𝐹𝑉)) ⊆ 𝑡
98 resabs1 6012 . . . . . . . . . . . . 13 ((𝑡 ∩ (𝐹𝑉)) ⊆ 𝑡 → ((𝐹𝑡) ↾ (𝑡 ∩ (𝐹𝑉))) = (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))))
9997, 98ax-mp 5 . . . . . . . . . . . 12 ((𝐹𝑡) ↾ (𝑡 ∩ (𝐹𝑉))) = (𝐹 ↾ (𝑡 ∩ (𝐹𝑉)))
1007cvmshmeo 35012 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝑆𝑈) ∧ 𝑡𝑥) → (𝐹𝑡) ∈ ((𝐶t 𝑡)Homeo(𝐽t 𝑈)))
101100adantll 712 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝐹𝑡) ∈ ((𝐶t 𝑡)Homeo(𝐽t 𝑈)))
1025adantr 479 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝐶 ∈ Top)
1039sselda 3976 . . . . . . . . . . . . . . . 16 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝑡𝐶)
104 elssuni 4941 . . . . . . . . . . . . . . . 16 (𝑡𝐶𝑡 𝐶)
105103, 104syl 17 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝑡 𝐶)
106 eqid 2725 . . . . . . . . . . . . . . . 16 𝐶 = 𝐶
107106restuni 23110 . . . . . . . . . . . . . . 15 ((𝐶 ∈ Top ∧ 𝑡 𝐶) → 𝑡 = (𝐶t 𝑡))
108102, 105, 107syl2anc 582 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝑡 = (𝐶t 𝑡))
10997, 108sseqtrid 4029 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝑡 ∩ (𝐹𝑉)) ⊆ (𝐶t 𝑡))
110 eqid 2725 . . . . . . . . . . . . . 14 (𝐶t 𝑡) = (𝐶t 𝑡)
111110hmeores 23719 . . . . . . . . . . . . 13 (((𝐹𝑡) ∈ ((𝐶t 𝑡)Homeo(𝐽t 𝑈)) ∧ (𝑡 ∩ (𝐹𝑉)) ⊆ (𝐶t 𝑡)) → ((𝐹𝑡) ↾ (𝑡 ∩ (𝐹𝑉))) ∈ (((𝐶t 𝑡) ↾t (𝑡 ∩ (𝐹𝑉)))Homeo((𝐽t 𝑈) ↾t ((𝐹𝑡) “ (𝑡 ∩ (𝐹𝑉))))))
112101, 109, 111syl2anc 582 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐹𝑡) ↾ (𝑡 ∩ (𝐹𝑉))) ∈ (((𝐶t 𝑡) ↾t (𝑡 ∩ (𝐹𝑉)))Homeo((𝐽t 𝑈) ↾t ((𝐹𝑡) “ (𝑡 ∩ (𝐹𝑉))))))
11399, 112eqeltrrid 2830 . . . . . . . . . . 11 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))) ∈ (((𝐶t 𝑡) ↾t (𝑡 ∩ (𝐹𝑉)))Homeo((𝐽t 𝑈) ↾t ((𝐹𝑡) “ (𝑡 ∩ (𝐹𝑉))))))
11497a1i 11 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝑡 ∩ (𝐹𝑉)) ⊆ 𝑡)
115 simpr 483 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝑡𝑥)
116 restabs 23113 . . . . . . . . . . . . 13 ((𝐶 ∈ Top ∧ (𝑡 ∩ (𝐹𝑉)) ⊆ 𝑡𝑡𝑥) → ((𝐶t 𝑡) ↾t (𝑡 ∩ (𝐹𝑉))) = (𝐶t (𝑡 ∩ (𝐹𝑉))))
117102, 114, 115, 116syl3anc 1368 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐶t 𝑡) ↾t (𝑡 ∩ (𝐹𝑉))) = (𝐶t (𝑡 ∩ (𝐹𝑉))))
118 incom 4199 . . . . . . . . . . . . . . . . 17 (𝑡 ∩ (𝐹𝑉)) = ((𝐹𝑉) ∩ 𝑡)
119 cnvresima 6236 . . . . . . . . . . . . . . . . 17 ((𝐹𝑡) “ 𝑉) = ((𝐹𝑉) ∩ 𝑡)
120118, 119eqtr4i 2756 . . . . . . . . . . . . . . . 16 (𝑡 ∩ (𝐹𝑉)) = ((𝐹𝑡) “ 𝑉)
121120imaeq2i 6062 . . . . . . . . . . . . . . 15 ((𝐹𝑡) “ (𝑡 ∩ (𝐹𝑉))) = ((𝐹𝑡) “ ((𝐹𝑡) “ 𝑉))
1223adantr 479 . . . . . . . . . . . . . . . . . 18 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝐹 ∈ (𝐶 CovMap 𝐽))
123 simplr 767 . . . . . . . . . . . . . . . . . 18 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝑥 ∈ (𝑆𝑈))
1247cvmsf1o 35013 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥 ∈ (𝑆𝑈) ∧ 𝑡𝑥) → (𝐹𝑡):𝑡1-1-onto𝑈)
125122, 123, 115, 124syl3anc 1368 . . . . . . . . . . . . . . . . 17 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝐹𝑡):𝑡1-1-onto𝑈)
126 f1ofo 6845 . . . . . . . . . . . . . . . . 17 ((𝐹𝑡):𝑡1-1-onto𝑈 → (𝐹𝑡):𝑡onto𝑈)
127125, 126syl 17 . . . . . . . . . . . . . . . 16 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝐹𝑡):𝑡onto𝑈)
12839adantr 479 . . . . . . . . . . . . . . . 16 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝑉𝑈)
129 foimacnv 6855 . . . . . . . . . . . . . . . 16 (((𝐹𝑡):𝑡onto𝑈𝑉𝑈) → ((𝐹𝑡) “ ((𝐹𝑡) “ 𝑉)) = 𝑉)
130127, 128, 129syl2anc 582 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐹𝑡) “ ((𝐹𝑡) “ 𝑉)) = 𝑉)
131121, 130eqtrid 2777 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐹𝑡) “ (𝑡 ∩ (𝐹𝑉))) = 𝑉)
132131oveq2d 7435 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐽t 𝑈) ↾t ((𝐹𝑡) “ (𝑡 ∩ (𝐹𝑉)))) = ((𝐽t 𝑈) ↾t 𝑉))
133 cvmtop2 35002 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
1343, 133syl 17 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝐽 ∈ Top)
1357cvmsrcl 35005 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑆𝑈) → 𝑈𝐽)
136135adantl 480 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝑈𝐽)
137 restabs 23113 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑉𝑈𝑈𝐽) → ((𝐽t 𝑈) ↾t 𝑉) = (𝐽t 𝑉))
138134, 39, 136, 137syl3anc 1368 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ((𝐽t 𝑈) ↾t 𝑉) = (𝐽t 𝑉))
139138adantr 479 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐽t 𝑈) ↾t 𝑉) = (𝐽t 𝑉))
140132, 139eqtrd 2765 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐽t 𝑈) ↾t ((𝐹𝑡) “ (𝑡 ∩ (𝐹𝑉)))) = (𝐽t 𝑉))
141117, 140oveq12d 7437 . . . . . . . . . . 11 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (((𝐶t 𝑡) ↾t (𝑡 ∩ (𝐹𝑉)))Homeo((𝐽t 𝑈) ↾t ((𝐹𝑡) “ (𝑡 ∩ (𝐹𝑉))))) = ((𝐶t (𝑡 ∩ (𝐹𝑉)))Homeo(𝐽t 𝑉)))
142113, 141eleqtrd 2827 . . . . . . . . . 10 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))) ∈ ((𝐶t (𝑡 ∩ (𝐹𝑉)))Homeo(𝐽t 𝑉)))
14396, 142jca 510 . . . . . . . . 9 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))})((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅ ∧ (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))) ∈ ((𝐶t (𝑡 ∩ (𝐹𝑉)))Homeo(𝐽t 𝑉))))
144143ralrimiva 3135 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ∀𝑡𝑥 (∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))})((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅ ∧ (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))) ∈ ((𝐶t (𝑡 ∩ (𝐹𝑉)))Homeo(𝐽t 𝑉))))
14552rgenw 3054 . . . . . . . . 9 𝑡𝑥 (𝑡 ∩ (𝐹𝑉)) ∈ V
14647cbvmptv 5262 . . . . . . . . . 10 (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = (𝑡𝑥 ↦ (𝑡 ∩ (𝐹𝑉)))
147 sneq 4640 . . . . . . . . . . . . 13 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → {𝑤} = {(𝑡 ∩ (𝐹𝑉))})
148147difeq2d 4118 . . . . . . . . . . . 12 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {𝑤}) = (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))}))
149 ineq1 4203 . . . . . . . . . . . . 13 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → (𝑤𝑧) = ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧))
150149eqeq1d 2727 . . . . . . . . . . . 12 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → ((𝑤𝑧) = ∅ ↔ ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅))
151148, 150raleqbidv 3329 . . . . . . . . . . 11 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → (∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {𝑤})(𝑤𝑧) = ∅ ↔ ∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))})((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅))
152 reseq2 5980 . . . . . . . . . . . 12 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → (𝐹𝑤) = (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))))
153 oveq2 7427 . . . . . . . . . . . . 13 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → (𝐶t 𝑤) = (𝐶t (𝑡 ∩ (𝐹𝑉))))
154153oveq1d 7434 . . . . . . . . . . . 12 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → ((𝐶t 𝑤)Homeo(𝐽t 𝑉)) = ((𝐶t (𝑡 ∩ (𝐹𝑉)))Homeo(𝐽t 𝑉)))
155152, 154eleq12d 2819 . . . . . . . . . . 11 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → ((𝐹𝑤) ∈ ((𝐶t 𝑤)Homeo(𝐽t 𝑉)) ↔ (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))) ∈ ((𝐶t (𝑡 ∩ (𝐹𝑉)))Homeo(𝐽t 𝑉))))
156151, 155anbi12d 630 . . . . . . . . . 10 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → ((∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {𝑤})(𝑤𝑧) = ∅ ∧ (𝐹𝑤) ∈ ((𝐶t 𝑤)Homeo(𝐽t 𝑉))) ↔ (∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))})((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅ ∧ (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))) ∈ ((𝐶t (𝑡 ∩ (𝐹𝑉)))Homeo(𝐽t 𝑉)))))
157146, 156ralrnmptw 7103 . . . . . . . . 9 (∀𝑡𝑥 (𝑡 ∩ (𝐹𝑉)) ∈ V → (∀𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))(∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {𝑤})(𝑤𝑧) = ∅ ∧ (𝐹𝑤) ∈ ((𝐶t 𝑤)Homeo(𝐽t 𝑉))) ↔ ∀𝑡𝑥 (∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))})((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅ ∧ (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))) ∈ ((𝐶t (𝑡 ∩ (𝐹𝑉)))Homeo(𝐽t 𝑉)))))
158145, 157ax-mp 5 . . . . . . . 8 (∀𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))(∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {𝑤})(𝑤𝑧) = ∅ ∧ (𝐹𝑤) ∈ ((𝐶t 𝑤)Homeo(𝐽t 𝑉))) ↔ ∀𝑡𝑥 (∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))})((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅ ∧ (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))) ∈ ((𝐶t (𝑡 ∩ (𝐹𝑉)))Homeo(𝐽t 𝑉))))
159144, 158sylibr 233 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ∀𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))(∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {𝑤})(𝑤𝑧) = ∅ ∧ (𝐹𝑤) ∈ ((𝐶t 𝑤)Homeo(𝐽t 𝑉))))
16068, 159jca 510 . . . . . 6 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ( ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = (𝐹𝑉) ∧ ∀𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))(∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {𝑤})(𝑤𝑧) = ∅ ∧ (𝐹𝑤) ∈ ((𝐶t 𝑤)Homeo(𝐽t 𝑉)))))
1617cvmscbv 34999 . . . . . . . 8 𝑆 = (𝑎𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑏 = (𝐹𝑎) ∧ ∀𝑤𝑏 (∀𝑧 ∈ (𝑏 ∖ {𝑤})(𝑤𝑧) = ∅ ∧ (𝐹𝑤) ∈ ((𝐶t 𝑤)Homeo(𝐽t 𝑎))))})
162161cvmsval 35007 . . . . . . 7 (𝐶 ∈ Top → (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∈ (𝑆𝑉) ↔ (𝑉𝐽 ∧ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ 𝐶 ∧ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ≠ ∅) ∧ ( ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = (𝐹𝑉) ∧ ∀𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))(∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {𝑤})(𝑤𝑧) = ∅ ∧ (𝐹𝑤) ∈ ((𝐶t 𝑤)Homeo(𝐽t 𝑉)))))))
1635, 162syl 17 . . . . . 6 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∈ (𝑆𝑉) ↔ (𝑉𝐽 ∧ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ 𝐶 ∧ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ≠ ∅) ∧ ( ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = (𝐹𝑉) ∧ ∀𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))(∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {𝑤})(𝑤𝑧) = ∅ ∧ (𝐹𝑤) ∈ ((𝐶t 𝑤)Homeo(𝐽t 𝑉)))))))
1642, 30, 160, 163mpbir3and 1339 . . . . 5 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∈ (𝑆𝑉))
165164ne0d 4335 . . . 4 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (𝑆𝑉) ≠ ∅)
166165ex 411 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) → (𝑥 ∈ (𝑆𝑈) → (𝑆𝑉) ≠ ∅))
167166exlimdv 1928 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) → (∃𝑥 𝑥 ∈ (𝑆𝑈) → (𝑆𝑉) ≠ ∅))
1681, 167biimtrid 241 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) → ((𝑆𝑈) ≠ ∅ → (𝑆𝑉) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wne 2929  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  cdif 3941  cin 3943  wss 3944  c0 4322  𝒫 cpw 4604  {csn 4630   cuni 4909  cmpt 5232  ccnv 5677  dom cdm 5678  ran crn 5679  cres 5680  cima 5681  ontowfo 6547  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  t crest 17405  Topctop 22839   Cn ccn 23172  Homeochmeo 23701   CovMap ccvm 34996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-map 8847  df-en 8965  df-fin 8968  df-fi 9436  df-rest 17407  df-topgen 17428  df-top 22840  df-topon 22857  df-bases 22893  df-cn 23175  df-hmeo 23703  df-cvm 34997
This theorem is referenced by:  cvmcov2  35016
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