| Step | Hyp | Ref
| Expression |
| 1 | | simp1 1137 |
. . 3
⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
| 2 | | simp3 1139 |
. . . 4
⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → 𝑃 ∈ 𝑈) |
| 3 | | simp2 1138 |
. . . 4
⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → 𝑈 ∈ 𝐽) |
| 4 | | elunii 4912 |
. . . 4
⊢ ((𝑃 ∈ 𝑈 ∧ 𝑈 ∈ 𝐽) → 𝑃 ∈ ∪ 𝐽) |
| 5 | 2, 3, 4 | syl2anc 584 |
. . 3
⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → 𝑃 ∈ ∪ 𝐽) |
| 6 | | cvmcov.1 |
. . . 4
⊢ 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))))}) |
| 7 | | eqid 2737 |
. . . 4
⊢ ∪ 𝐽 =
∪ 𝐽 |
| 8 | 6, 7 | cvmcov 35268 |
. . 3
⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃 ∈ ∪ 𝐽) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅)) |
| 9 | 1, 5, 8 | syl2anc 584 |
. 2
⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅)) |
| 10 | | inss2 4238 |
. . . . 5
⊢ (𝑦 ∩ 𝑈) ⊆ 𝑈 |
| 11 | | vex 3484 |
. . . . . . 7
⊢ 𝑦 ∈ V |
| 12 | 11 | inex1 5317 |
. . . . . 6
⊢ (𝑦 ∩ 𝑈) ∈ V |
| 13 | 12 | elpw 4604 |
. . . . 5
⊢ ((𝑦 ∩ 𝑈) ∈ 𝒫 𝑈 ↔ (𝑦 ∩ 𝑈) ⊆ 𝑈) |
| 14 | 10, 13 | mpbir 231 |
. . . 4
⊢ (𝑦 ∩ 𝑈) ∈ 𝒫 𝑈 |
| 15 | 14 | a1i 11 |
. . 3
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → (𝑦 ∩ 𝑈) ∈ 𝒫 𝑈) |
| 16 | | simprrl 781 |
. . . 4
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝑃 ∈ 𝑦) |
| 17 | 2 | adantr 480 |
. . . 4
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝑃 ∈ 𝑈) |
| 18 | 16, 17 | elind 4200 |
. . 3
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝑃 ∈ (𝑦 ∩ 𝑈)) |
| 19 | | simprrr 782 |
. . . 4
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → (𝑆‘𝑦) ≠ ∅) |
| 20 | 1 | adantr 480 |
. . . . 5
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
| 21 | | cvmtop2 35266 |
. . . . . . 7
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top) |
| 22 | 20, 21 | syl 17 |
. . . . . 6
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝐽 ∈ Top) |
| 23 | | simprl 771 |
. . . . . 6
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝑦 ∈ 𝐽) |
| 24 | 3 | adantr 480 |
. . . . . 6
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝑈 ∈ 𝐽) |
| 25 | | inopn 22905 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝑈 ∈ 𝐽) → (𝑦 ∩ 𝑈) ∈ 𝐽) |
| 26 | 22, 23, 24, 25 | syl3anc 1373 |
. . . . 5
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → (𝑦 ∩ 𝑈) ∈ 𝐽) |
| 27 | | inss1 4237 |
. . . . . 6
⊢ (𝑦 ∩ 𝑈) ⊆ 𝑦 |
| 28 | 27 | a1i 11 |
. . . . 5
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → (𝑦 ∩ 𝑈) ⊆ 𝑦) |
| 29 | 6 | cvmsss2 35279 |
. . . . 5
⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑦 ∩ 𝑈) ∈ 𝐽 ∧ (𝑦 ∩ 𝑈) ⊆ 𝑦) → ((𝑆‘𝑦) ≠ ∅ → (𝑆‘(𝑦 ∩ 𝑈)) ≠ ∅)) |
| 30 | 20, 26, 28, 29 | syl3anc 1373 |
. . . 4
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → ((𝑆‘𝑦) ≠ ∅ → (𝑆‘(𝑦 ∩ 𝑈)) ≠ ∅)) |
| 31 | 19, 30 | mpd 15 |
. . 3
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → (𝑆‘(𝑦 ∩ 𝑈)) ≠ ∅) |
| 32 | | eleq2 2830 |
. . . . 5
⊢ (𝑥 = (𝑦 ∩ 𝑈) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑦 ∩ 𝑈))) |
| 33 | | fveq2 6906 |
. . . . . 6
⊢ (𝑥 = (𝑦 ∩ 𝑈) → (𝑆‘𝑥) = (𝑆‘(𝑦 ∩ 𝑈))) |
| 34 | 33 | neeq1d 3000 |
. . . . 5
⊢ (𝑥 = (𝑦 ∩ 𝑈) → ((𝑆‘𝑥) ≠ ∅ ↔ (𝑆‘(𝑦 ∩ 𝑈)) ≠ ∅)) |
| 35 | 32, 34 | anbi12d 632 |
. . . 4
⊢ (𝑥 = (𝑦 ∩ 𝑈) → ((𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅) ↔ (𝑃 ∈ (𝑦 ∩ 𝑈) ∧ (𝑆‘(𝑦 ∩ 𝑈)) ≠ ∅))) |
| 36 | 35 | rspcev 3622 |
. . 3
⊢ (((𝑦 ∩ 𝑈) ∈ 𝒫 𝑈 ∧ (𝑃 ∈ (𝑦 ∩ 𝑈) ∧ (𝑆‘(𝑦 ∩ 𝑈)) ≠ ∅)) → ∃𝑥 ∈ 𝒫 𝑈(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅)) |
| 37 | 15, 18, 31, 36 | syl12anc 837 |
. 2
⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → ∃𝑥 ∈ 𝒫 𝑈(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅)) |
| 38 | 9, 37 | rexlimddv 3161 |
1
⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑥 ∈ 𝒫 𝑈(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅)) |