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Theorem cvmsss2 31704
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 4095 . 2 ((𝑆𝑈) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆𝑈))
2 simpl2 1244 . . . . . 6 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝑉𝐽)
3 simpl1 1242 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
4 cvmtop1 31690 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
53, 4syl 17 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝐶 ∈ Top)
65adantr 472 . . . . . . . . . 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 31697 . . . . . . . . . . . 12 (𝑥 ∈ (𝑆𝑈) → 𝑥𝐶)
98adantl 473 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝑥𝐶)
109sselda 3761 . . . . . . . . . 10 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑦𝑥) → 𝑦𝐶)
11 cvmcn 31692 . . . . . . . . . . . . 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 21349 . . . . . . . . . . . 12 ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝑉𝐽) → (𝐹𝑉) ∈ 𝐶)
1412, 2, 13syl2anc 579 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (𝐹𝑉) ∈ 𝐶)
1514adantr 472 . . . . . . . . . 10 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑦𝑥) → (𝐹𝑉) ∈ 𝐶)
16 inopn 20983 . . . . . . . . . 10 ((𝐶 ∈ Top ∧ 𝑦𝐶 ∧ (𝐹𝑉) ∈ 𝐶) → (𝑦 ∩ (𝐹𝑉)) ∈ 𝐶)
176, 10, 15, 16syl3anc 1490 . . . . . . . . 9 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑦𝑥) → (𝑦 ∩ (𝐹𝑉)) ∈ 𝐶)
1817fmpttd 6575 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))):𝑥𝐶)
1918frnd 6230 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ 𝐶)
207cvmsn0 31698 . . . . . . . . 9 (𝑥 ∈ (𝑆𝑈) → 𝑥 ≠ ∅)
2120adantl 473 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝑥 ≠ ∅)
22 dmmptg 5818 . . . . . . . . . . . 12 (∀𝑦𝑥 (𝑦 ∩ (𝐹𝑉)) ∈ V → dom (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = 𝑥)
23 inex1g 4962 . . . . . . . . . . . 12 (𝑦𝑥 → (𝑦 ∩ (𝐹𝑉)) ∈ V)
2422, 23mprg 3073 . . . . . . . . . . 11 dom (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = 𝑥
2524eqeq1i 2770 . . . . . . . . . 10 (dom (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = ∅ ↔ 𝑥 = ∅)
26 dm0rn0 5510 . . . . . . . . . 10 (dom (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = ∅ ↔ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = ∅)
2725, 26bitr3i 268 . . . . . . . . 9 (𝑥 = ∅ ↔ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = ∅)
2827necon3bii 2989 . . . . . . . 8 (𝑥 ≠ ∅ ↔ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ≠ ∅)
2921, 28sylib 209 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ≠ ∅)
3019, 29jca 507 . . . . . 6 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ 𝐶 ∧ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ≠ ∅))
31 inss2 3993 . . . . . . . . . . . 12 (𝑦 ∩ (𝐹𝑉)) ⊆ (𝐹𝑉)
32 elpw2g 4985 . . . . . . . . . . . . 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 249 . . . . . . . . . . 11 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑦𝑥) → (𝑦 ∩ (𝐹𝑉)) ∈ 𝒫 (𝐹𝑉))
3534fmpttd 6575 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))):𝑥⟶𝒫 (𝐹𝑉))
3635frnd 6230 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ 𝒫 (𝐹𝑉))
37 sspwuni 4768 . . . . . . . . 9 (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ 𝒫 (𝐹𝑉) ↔ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ (𝐹𝑉))
3836, 37sylib 209 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ⊆ (𝐹𝑉))
39 simpl3 1246 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝑉𝑈)
40 imass2 5683 . . . . . . . . . . . 12 (𝑉𝑈 → (𝐹𝑉) ⊆ (𝐹𝑈))
4139, 40syl 17 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (𝐹𝑉) ⊆ (𝐹𝑈))
427cvmsuni 31699 . . . . . . . . . . . 12 (𝑥 ∈ (𝑆𝑈) → 𝑥 = (𝐹𝑈))
4342adantl 473 . . . . . . . . . . 11 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝑥 = (𝐹𝑈))
4441, 43sseqtr4d 3802 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (𝐹𝑉) ⊆ 𝑥)
4544sselda 3761 . . . . . . . . 9 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) → 𝑧 𝑥)
46 eqid 2765 . . . . . . . . . . . . . . 15 (𝑡 ∩ (𝐹𝑉)) = (𝑡 ∩ (𝐹𝑉))
47 ineq1 3969 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑡 → (𝑦 ∩ (𝐹𝑉)) = (𝑡 ∩ (𝐹𝑉)))
4847rspceeqv 3479 . . . . . . . . . . . . . . 15 ((𝑡𝑥 ∧ (𝑡 ∩ (𝐹𝑉)) = (𝑡 ∩ (𝐹𝑉))) → ∃𝑦𝑥 (𝑡 ∩ (𝐹𝑉)) = (𝑦 ∩ (𝐹𝑉)))
4946, 48mpan2 682 . . . . . . . . . . . . . 14 (𝑡𝑥 → ∃𝑦𝑥 (𝑡 ∩ (𝐹𝑉)) = (𝑦 ∩ (𝐹𝑉)))
5049ad2antrl 719 . . . . . . . . . . . . 13 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) ∧ (𝑡𝑥𝑧𝑡)) → ∃𝑦𝑥 (𝑡 ∩ (𝐹𝑉)) = (𝑦 ∩ (𝐹𝑉)))
51 vex 3353 . . . . . . . . . . . . . . 15 𝑡 ∈ V
5251inex1 4960 . . . . . . . . . . . . . 14 (𝑡 ∩ (𝐹𝑉)) ∈ V
53 eqid 2765 . . . . . . . . . . . . . . 15 (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))
5453elrnmpt 5541 . . . . . . . . . . . . . 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 225 . . . . . . . . . . . 12 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) ∧ (𝑡𝑥𝑧𝑡)) → (𝑡 ∩ (𝐹𝑉)) ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))))
57 simprr 789 . . . . . . . . . . . . 13 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) ∧ (𝑡𝑥𝑧𝑡)) → 𝑧𝑡)
58 simplr 785 . . . . . . . . . . . . 13 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) ∧ (𝑡𝑥𝑧𝑡)) → 𝑧 ∈ (𝐹𝑉))
5957, 58elind 3960 . . . . . . . . . . . 12 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) ∧ (𝑡𝑥𝑧𝑡)) → 𝑧 ∈ (𝑡 ∩ (𝐹𝑉)))
60 eleq2 2833 . . . . . . . . . . . . 13 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → (𝑧𝑤𝑧 ∈ (𝑡 ∩ (𝐹𝑉))))
6160rspcev 3461 . . . . . . . . . . . 12 (((𝑡 ∩ (𝐹𝑉)) ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∧ 𝑧 ∈ (𝑡 ∩ (𝐹𝑉))) → ∃𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))𝑧𝑤)
6256, 59, 61syl2anc 579 . . . . . . . . . . 11 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) ∧ (𝑡𝑥𝑧𝑡)) → ∃𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))𝑧𝑤)
6362rexlimdvaa 3179 . . . . . . . . . 10 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑧 ∈ (𝐹𝑉)) → (∃𝑡𝑥 𝑧𝑡 → ∃𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))𝑧𝑤))
64 eluni2 4598 . . . . . . . . . 10 (𝑧 𝑥 ↔ ∃𝑡𝑥 𝑧𝑡)
65 eluni2 4598 . . . . . . . . . 10 (𝑧 ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ↔ ∃𝑤 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉)))𝑧𝑤)
6663, 64, 653imtr4g 287 . . . . . . . . 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 3782 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = (𝐹𝑉))
69 eldifsn 4472 . . . . . . . . . . . 12 (𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))}) ↔ (𝑧 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∧ 𝑧 ≠ (𝑡 ∩ (𝐹𝑉))))
70 vex 3353 . . . . . . . . . . . . . . 15 𝑧 ∈ V
7153elrnmpt 5541 . . . . . . . . . . . . . . 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 2117 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑦 → (𝑦 ∩ (𝐹𝑉)) = (𝑡 ∩ (𝐹𝑉)))
7473necon3ai 2962 . . . . . . . . . . . . . . . . 17 ((𝑦 ∩ (𝐹𝑉)) ≠ (𝑡 ∩ (𝐹𝑉)) → ¬ 𝑡 = 𝑦)
75 simpllr 793 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → 𝑥 ∈ (𝑆𝑈))
76 simplr 785 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → 𝑡𝑥)
77 simpr 477 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → 𝑦𝑥)
787cvmsdisj 31700 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ (𝑆𝑈) ∧ 𝑡𝑥𝑦𝑥) → (𝑡 = 𝑦 ∨ (𝑡𝑦) = ∅))
7975, 76, 77, 78syl3anc 1490 . . . . . . . . . . . . . . . . . 18 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → (𝑡 = 𝑦 ∨ (𝑡𝑦) = ∅))
8079ord 890 . . . . . . . . . . . . . . . . 17 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → (¬ 𝑡 = 𝑦 → (𝑡𝑦) = ∅))
81 inss1 3992 . . . . . . . . . . . . . . . . . 18 ((𝑡𝑦) ∩ (𝐹𝑉)) ⊆ (𝑡𝑦)
82 sseq0 4137 . . . . . . . . . . . . . . . . . 18 ((((𝑡𝑦) ∩ (𝐹𝑉)) ⊆ (𝑡𝑦) ∧ (𝑡𝑦) = ∅) → ((𝑡𝑦) ∩ (𝐹𝑉)) = ∅)
8381, 82mpan 681 . . . . . . . . . . . . . . . . 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 2999 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑦 ∩ (𝐹𝑉)) → (𝑧 ≠ (𝑡 ∩ (𝐹𝑉)) ↔ (𝑦 ∩ (𝐹𝑉)) ≠ (𝑡 ∩ (𝐹𝑉))))
86 ineq2 3970 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑦 ∩ (𝐹𝑉)) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ((𝑡 ∩ (𝐹𝑉)) ∩ (𝑦 ∩ (𝐹𝑉))))
87 inindir 3991 . . . . . . . . . . . . . . . . . . 19 ((𝑡𝑦) ∩ (𝐹𝑉)) = ((𝑡 ∩ (𝐹𝑉)) ∩ (𝑦 ∩ (𝐹𝑉)))
8886, 87syl6eqr 2817 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑦 ∩ (𝐹𝑉)) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ((𝑡𝑦) ∩ (𝐹𝑉)))
8988eqeq1d 2767 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑦 ∩ (𝐹𝑉)) → (((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅ ↔ ((𝑡𝑦) ∩ (𝐹𝑉)) = ∅))
9085, 89imbi12d 335 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑦 ∩ (𝐹𝑉)) → ((𝑧 ≠ (𝑡 ∩ (𝐹𝑉)) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅) ↔ ((𝑦 ∩ (𝐹𝑉)) ≠ (𝑡 ∩ (𝐹𝑉)) → ((𝑡𝑦) ∩ (𝐹𝑉)) = ∅)))
9184, 90syl5ibrcom 238 . . . . . . . . . . . . . . 15 (((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) ∧ 𝑦𝑥) → (𝑧 = (𝑦 ∩ (𝐹𝑉)) → (𝑧 ≠ (𝑡 ∩ (𝐹𝑉)) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅)))
9291rexlimdva 3178 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (∃𝑦𝑥 𝑧 = (𝑦 ∩ (𝐹𝑉)) → (𝑧 ≠ (𝑡 ∩ (𝐹𝑉)) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅)))
9372, 92syl5bi 233 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝑧 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) → (𝑧 ≠ (𝑡 ∩ (𝐹𝑉)) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅)))
9493impd 398 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝑧 ∈ ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∧ 𝑧 ≠ (𝑡 ∩ (𝐹𝑉))) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅))
9569, 94syl5bi 233 . . . . . . . . . . 11 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))}) → ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅))
9695ralrimiv 3112 . . . . . . . . . 10 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))})((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅)
97 inss1 3992 . . . . . . . . . . . . 13 (𝑡 ∩ (𝐹𝑉)) ⊆ 𝑡
98 resabs1 5602 . . . . . . . . . . . . 13 ((𝑡 ∩ (𝐹𝑉)) ⊆ 𝑡 → ((𝐹𝑡) ↾ (𝑡 ∩ (𝐹𝑉))) = (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))))
9997, 98ax-mp 5 . . . . . . . . . . . 12 ((𝐹𝑡) ↾ (𝑡 ∩ (𝐹𝑉))) = (𝐹 ↾ (𝑡 ∩ (𝐹𝑉)))
1007cvmshmeo 31701 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝑆𝑈) ∧ 𝑡𝑥) → (𝐹𝑡) ∈ ((𝐶t 𝑡)Homeo(𝐽t 𝑈)))
101100adantll 705 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝐹𝑡) ∈ ((𝐶t 𝑡)Homeo(𝐽t 𝑈)))
1025adantr 472 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝐶 ∈ Top)
1039sselda 3761 . . . . . . . . . . . . . . . 16 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝑡𝐶)
104 elssuni 4625 . . . . . . . . . . . . . . . 16 (𝑡𝐶𝑡 𝐶)
105103, 104syl 17 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝑡 𝐶)
106 eqid 2765 . . . . . . . . . . . . . . . 16 𝐶 = 𝐶
107106restuni 21246 . . . . . . . . . . . . . . 15 ((𝐶 ∈ Top ∧ 𝑡 𝐶) → 𝑡 = (𝐶t 𝑡))
108102, 105, 107syl2anc 579 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝑡 = (𝐶t 𝑡))
10997, 108syl5sseq 3813 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝑡 ∩ (𝐹𝑉)) ⊆ (𝐶t 𝑡))
110 eqid 2765 . . . . . . . . . . . . . 14 (𝐶t 𝑡) = (𝐶t 𝑡)
111110hmeores 21854 . . . . . . . . . . . . 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 579 . . . . . . . . . . . 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, 112syl5eqelr 2849 . . . . . . . . . . 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 477 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝑡𝑥)
116 restabs 21249 . . . . . . . . . . . . 13 ((𝐶 ∈ Top ∧ (𝑡 ∩ (𝐹𝑉)) ⊆ 𝑡𝑡𝑥) → ((𝐶t 𝑡) ↾t (𝑡 ∩ (𝐹𝑉))) = (𝐶t (𝑡 ∩ (𝐹𝑉))))
117102, 114, 115, 116syl3anc 1490 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐶t 𝑡) ↾t (𝑡 ∩ (𝐹𝑉))) = (𝐶t (𝑡 ∩ (𝐹𝑉))))
118 incom 3967 . . . . . . . . . . . . . . . . 17 (𝑡 ∩ (𝐹𝑉)) = ((𝐹𝑉) ∩ 𝑡)
119 cnvresima 5809 . . . . . . . . . . . . . . . . 17 ((𝐹𝑡) “ 𝑉) = ((𝐹𝑉) ∩ 𝑡)
120118, 119eqtr4i 2790 . . . . . . . . . . . . . . . 16 (𝑡 ∩ (𝐹𝑉)) = ((𝐹𝑡) “ 𝑉)
121120imaeq2i 5646 . . . . . . . . . . . . . . 15 ((𝐹𝑡) “ (𝑡 ∩ (𝐹𝑉))) = ((𝐹𝑡) “ ((𝐹𝑡) “ 𝑉))
1223adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝐹 ∈ (𝐶 CovMap 𝐽))
123 simplr 785 . . . . . . . . . . . . . . . . . 18 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝑥 ∈ (𝑆𝑈))
1247cvmsf1o 31702 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥 ∈ (𝑆𝑈) ∧ 𝑡𝑥) → (𝐹𝑡):𝑡1-1-onto𝑈)
125122, 123, 115, 124syl3anc 1490 . . . . . . . . . . . . . . . . 17 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝐹𝑡):𝑡1-1-onto𝑈)
126 f1ofo 6327 . . . . . . . . . . . . . . . . 17 ((𝐹𝑡):𝑡1-1-onto𝑈 → (𝐹𝑡):𝑡onto𝑈)
127125, 126syl 17 . . . . . . . . . . . . . . . 16 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝐹𝑡):𝑡onto𝑈)
12839adantr 472 . . . . . . . . . . . . . . . 16 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → 𝑉𝑈)
129 foimacnv 6337 . . . . . . . . . . . . . . . 16 (((𝐹𝑡):𝑡onto𝑈𝑉𝑈) → ((𝐹𝑡) “ ((𝐹𝑡) “ 𝑉)) = 𝑉)
130127, 128, 129syl2anc 579 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐹𝑡) “ ((𝐹𝑡) “ 𝑉)) = 𝑉)
131121, 130syl5eq 2811 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐹𝑡) “ (𝑡 ∩ (𝐹𝑉))) = 𝑉)
132131oveq2d 6858 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐽t 𝑈) ↾t ((𝐹𝑡) “ (𝑡 ∩ (𝐹𝑉)))) = ((𝐽t 𝑈) ↾t 𝑉))
133 cvmtop2 31691 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
1343, 133syl 17 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝐽 ∈ Top)
1357cvmsrcl 31694 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑆𝑈) → 𝑈𝐽)
136135adantl 473 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → 𝑈𝐽)
137 restabs 21249 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑉𝑈𝑈𝐽) → ((𝐽t 𝑈) ↾t 𝑉) = (𝐽t 𝑉))
138134, 39, 136, 137syl3anc 1490 . . . . . . . . . . . . . 14 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ((𝐽t 𝑈) ↾t 𝑉) = (𝐽t 𝑉))
139138adantr 472 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐽t 𝑈) ↾t 𝑉) = (𝐽t 𝑉))
140132, 139eqtrd 2799 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → ((𝐽t 𝑈) ↾t ((𝐹𝑡) “ (𝑡 ∩ (𝐹𝑉)))) = (𝐽t 𝑉))
141117, 140oveq12d 6860 . . . . . . . . . . 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 2846 . . . . . . . . . 10 ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) ∧ 𝑡𝑥) → (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))) ∈ ((𝐶t (𝑡 ∩ (𝐹𝑉)))Homeo(𝐽t 𝑉)))
14396, 142jca 507 . . . . . . . . 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 3113 . . . . . . . 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 3071 . . . . . . . . 9 𝑡𝑥 (𝑡 ∩ (𝐹𝑉)) ∈ V
14647cbvmptv 4909 . . . . . . . . . 10 (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) = (𝑡𝑥 ↦ (𝑡 ∩ (𝐹𝑉)))
147 sneq 4344 . . . . . . . . . . . . 13 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → {𝑤} = {(𝑡 ∩ (𝐹𝑉))})
148147difeq2d 3890 . . . . . . . . . . . 12 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {𝑤}) = (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))}))
149 ineq1 3969 . . . . . . . . . . . . 13 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → (𝑤𝑧) = ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧))
150149eqeq1d 2767 . . . . . . . . . . . 12 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → ((𝑤𝑧) = ∅ ↔ ((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅))
151148, 150raleqbidv 3300 . . . . . . . . . . 11 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → (∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {𝑤})(𝑤𝑧) = ∅ ↔ ∀𝑧 ∈ (ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∖ {(𝑡 ∩ (𝐹𝑉))})((𝑡 ∩ (𝐹𝑉)) ∩ 𝑧) = ∅))
152 reseq2 5560 . . . . . . . . . . . 12 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → (𝐹𝑤) = (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))))
153 oveq2 6850 . . . . . . . . . . . . 13 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → (𝐶t 𝑤) = (𝐶t (𝑡 ∩ (𝐹𝑉))))
154153oveq1d 6857 . . . . . . . . . . . 12 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → ((𝐶t 𝑤)Homeo(𝐽t 𝑉)) = ((𝐶t (𝑡 ∩ (𝐹𝑉)))Homeo(𝐽t 𝑉)))
155152, 154eleq12d 2838 . . . . . . . . . . 11 (𝑤 = (𝑡 ∩ (𝐹𝑉)) → ((𝐹𝑤) ∈ ((𝐶t 𝑤)Homeo(𝐽t 𝑉)) ↔ (𝐹 ↾ (𝑡 ∩ (𝐹𝑉))) ∈ ((𝐶t (𝑡 ∩ (𝐹𝑉)))Homeo(𝐽t 𝑉))))
156151, 155anbi12d 624 . . . . . . . . . 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, 156ralrnmpt 6558 . . . . . . . . 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 225 . . . . . . 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 507 . . . . . 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 31688 . . . . . . . 8 𝑆 = (𝑎𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑏 = (𝐹𝑎) ∧ ∀𝑤𝑏 (∀𝑧 ∈ (𝑏 ∖ {𝑤})(𝑤𝑧) = ∅ ∧ (𝐹𝑤) ∈ ((𝐶t 𝑤)Homeo(𝐽t 𝑎))))})
162161cvmsval 31696 . . . . . . 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 1442 . . . . 5 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → ran (𝑦𝑥 ↦ (𝑦 ∩ (𝐹𝑉))) ∈ (𝑆𝑉))
165164ne0d 4086 . . . 4 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) ∧ 𝑥 ∈ (𝑆𝑈)) → (𝑆𝑉) ≠ ∅)
166165ex 401 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) → (𝑥 ∈ (𝑆𝑈) → (𝑆𝑉) ≠ ∅))
167166exlimdv 2028 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) → (∃𝑥 𝑥 ∈ (𝑆𝑈) → (𝑆𝑉) ≠ ∅))
1681, 167syl5bi 233 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) → ((𝑆𝑈) ≠ ∅ → (𝑆𝑉) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3729  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334   cuni 4594  cmpt 4888  ccnv 5276  dom cdm 5277  ran crn 5278  cres 5279  cima 5280  ontowfo 6066  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  t crest 16347  Topctop 20977   Cn ccn 21308  Homeochmeo 21836   CovMap ccvm 31685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-fin 8164  df-fi 8524  df-rest 16349  df-topgen 16370  df-top 20978  df-topon 20995  df-bases 21030  df-cn 21311  df-hmeo 21838  df-cvm 31686
This theorem is referenced by:  cvmcov2  31705
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