Step | Hyp | Ref
| Expression |
1 | | cvmlift3.b |
. . 3
⊢ 𝐵 = ∪
𝐶 |
2 | | cvmlift3.y |
. . 3
⊢ 𝑌 = ∪
𝐾 |
3 | | cvmlift3.f |
. . 3
⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
4 | | cvmlift3.k |
. . 3
⊢ (𝜑 → 𝐾 ∈ SConn) |
5 | | cvmlift3.l |
. . 3
⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally
PConn) |
6 | | cvmlift3.o |
. . 3
⊢ (𝜑 → 𝑂 ∈ 𝑌) |
7 | | cvmlift3.g |
. . 3
⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽)) |
8 | | cvmlift3.p |
. . 3
⊢ (𝜑 → 𝑃 ∈ 𝐵) |
9 | | cvmlift3.e |
. . 3
⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂)) |
10 | | cvmlift3.h |
. . 3
⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cvmlift3lem3 33291 |
. 2
⊢ (𝜑 → 𝐻:𝑌⟶𝐵) |
12 | 3 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
13 | | eqid 2738 |
. . . . . . . 8
⊢ ∪ 𝐽 =
∪ 𝐽 |
14 | 2, 13 | cnf 22407 |
. . . . . . 7
⊢ (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌⟶∪ 𝐽) |
15 | 7, 14 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐺:𝑌⟶∪ 𝐽) |
16 | 15 | ffvelrnda 6953 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐺‘𝑦) ∈ ∪ 𝐽) |
17 | | cvmlift3lem7.s |
. . . . . 6
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 =
(◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))}) |
18 | 17, 13 | cvmcov 33233 |
. . . . 5
⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺‘𝑦) ∈ ∪ 𝐽) → ∃𝑎 ∈ 𝐽 ((𝐺‘𝑦) ∈ 𝑎 ∧ (𝑆‘𝑎) ≠ ∅)) |
19 | 12, 16, 18 | syl2anc 584 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∃𝑎 ∈ 𝐽 ((𝐺‘𝑦) ∈ 𝑎 ∧ (𝑆‘𝑎) ≠ ∅)) |
20 | | n0 4280 |
. . . . . . 7
⊢ ((𝑆‘𝑎) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ (𝑆‘𝑎)) |
21 | 5 | ad2antrr 723 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → 𝐾 ∈ 𝑛-Locally
PConn) |
22 | 7 | ad2antrr 723 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → 𝐺 ∈ (𝐾 Cn 𝐽)) |
23 | | simprr 770 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → 𝑡 ∈ (𝑆‘𝑎)) |
24 | 17 | cvmsrcl 33234 |
. . . . . . . . . . . . 13
⊢ (𝑡 ∈ (𝑆‘𝑎) → 𝑎 ∈ 𝐽) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → 𝑎 ∈ 𝐽) |
26 | | cnima 22426 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ (𝐾 Cn 𝐽) ∧ 𝑎 ∈ 𝐽) → (◡𝐺 “ 𝑎) ∈ 𝐾) |
27 | 22, 25, 26 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → (◡𝐺 “ 𝑎) ∈ 𝐾) |
28 | | simplr 766 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → 𝑦 ∈ 𝑌) |
29 | | simprl 768 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → (𝐺‘𝑦) ∈ 𝑎) |
30 | | ffn 6592 |
. . . . . . . . . . . . 13
⊢ (𝐺:𝑌⟶∪ 𝐽 → 𝐺 Fn 𝑌) |
31 | | elpreima 6927 |
. . . . . . . . . . . . 13
⊢ (𝐺 Fn 𝑌 → (𝑦 ∈ (◡𝐺 “ 𝑎) ↔ (𝑦 ∈ 𝑌 ∧ (𝐺‘𝑦) ∈ 𝑎))) |
32 | 22, 14, 30, 31 | 4syl 19 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → (𝑦 ∈ (◡𝐺 “ 𝑎) ↔ (𝑦 ∈ 𝑌 ∧ (𝐺‘𝑦) ∈ 𝑎))) |
33 | 28, 29, 32 | mpbir2and 710 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → 𝑦 ∈ (◡𝐺 “ 𝑎)) |
34 | | nlly2i 22637 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ 𝑛-Locally PConn
∧ (◡𝐺 “ 𝑎) ∈ 𝐾 ∧ 𝑦 ∈ (◡𝐺 “ 𝑎)) → ∃𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎)∃𝑣 ∈ 𝐾 (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn)) |
35 | 21, 27, 33, 34 | syl3anc 1370 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → ∃𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎)∃𝑣 ∈ 𝐾 (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn)) |
36 | 3 | ad3antrrr 727 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
37 | 4 | ad3antrrr 727 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝐾 ∈ SConn) |
38 | 5 | ad3antrrr 727 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝐾 ∈ 𝑛-Locally
PConn) |
39 | 6 | ad3antrrr 727 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝑂 ∈ 𝑌) |
40 | 7 | ad3antrrr 727 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝐺 ∈ (𝐾 Cn 𝐽)) |
41 | 8 | ad3antrrr 727 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝑃 ∈ 𝐵) |
42 | 9 | ad3antrrr 727 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → (𝐹‘𝑃) = (𝐺‘𝑂)) |
43 | 29 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → (𝐺‘𝑦) ∈ 𝑎) |
44 | 23 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝑡 ∈ (𝑆‘𝑎)) |
45 | | simprll 776 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎)) |
46 | 45 | elpwid 4544 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝑚 ⊆ (◡𝐺 “ 𝑎)) |
47 | | eqid 2738 |
. . . . . . . . . . . . 13
⊢
(℩𝑏
∈ 𝑡 (𝐻‘𝑦) ∈ 𝑏) = (℩𝑏 ∈ 𝑡 (𝐻‘𝑦) ∈ 𝑏) |
48 | | simprr3 1222 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → (𝐾 ↾t 𝑚) ∈ PConn) |
49 | | simprlr 777 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝑣 ∈ 𝐾) |
50 | | simprr2 1221 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝑣 ⊆ 𝑚) |
51 | | simprr1 1220 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝑦 ∈ 𝑣) |
52 | 1, 2, 36, 37, 38, 39, 40, 41, 42, 10, 17, 43, 44, 46, 47, 48, 49, 50, 51 | cvmlift3lem7 33295 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾) ∧ (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn))) → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦)) |
53 | 52 | expr 457 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎) ∧ 𝑣 ∈ 𝐾)) → ((𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn) → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦))) |
54 | 53 | rexlimdvva 3221 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → (∃𝑚 ∈ 𝒫 (◡𝐺 “ 𝑎)∃𝑣 ∈ 𝐾 (𝑦 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑚 ∧ (𝐾 ↾t 𝑚) ∈ PConn) → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦))) |
55 | 35, 54 | mpd 15 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ ((𝐺‘𝑦) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦)) |
56 | 55 | expr 457 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ (𝐺‘𝑦) ∈ 𝑎) → (𝑡 ∈ (𝑆‘𝑎) → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦))) |
57 | 56 | exlimdv 1936 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ (𝐺‘𝑦) ∈ 𝑎) → (∃𝑡 𝑡 ∈ (𝑆‘𝑎) → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦))) |
58 | 20, 57 | syl5bi 241 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ (𝐺‘𝑦) ∈ 𝑎) → ((𝑆‘𝑎) ≠ ∅ → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦))) |
59 | 58 | expimpd 454 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (((𝐺‘𝑦) ∈ 𝑎 ∧ (𝑆‘𝑎) ≠ ∅) → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦))) |
60 | 59 | rexlimdvw 3217 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (∃𝑎 ∈ 𝐽 ((𝐺‘𝑦) ∈ 𝑎 ∧ (𝑆‘𝑎) ≠ ∅) → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦))) |
61 | 19, 60 | mpd 15 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦)) |
62 | 61 | ralrimiva 3108 |
. 2
⊢ (𝜑 → ∀𝑦 ∈ 𝑌 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦)) |
63 | | sconntop 33198 |
. . . . 5
⊢ (𝐾 ∈ SConn → 𝐾 ∈ Top) |
64 | 4, 63 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ Top) |
65 | 2 | toptopon 22076 |
. . . 4
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌)) |
66 | 64, 65 | sylib 217 |
. . 3
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
67 | | cvmtop1 33230 |
. . . . 5
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top) |
68 | 3, 67 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ Top) |
69 | 1 | toptopon 22076 |
. . . 4
⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵)) |
70 | 68, 69 | sylib 217 |
. . 3
⊢ (𝜑 → 𝐶 ∈ (TopOn‘𝐵)) |
71 | | cncnp 22441 |
. . 3
⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐶 ∈ (TopOn‘𝐵)) → (𝐻 ∈ (𝐾 Cn 𝐶) ↔ (𝐻:𝑌⟶𝐵 ∧ ∀𝑦 ∈ 𝑌 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦)))) |
72 | 66, 70, 71 | syl2anc 584 |
. 2
⊢ (𝜑 → (𝐻 ∈ (𝐾 Cn 𝐶) ↔ (𝐻:𝑌⟶𝐵 ∧ ∀𝑦 ∈ 𝑌 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑦)))) |
73 | 11, 62, 72 | mpbir2and 710 |
1
⊢ (𝜑 → 𝐻 ∈ (𝐾 Cn 𝐶)) |