Proof of Theorem xkopjcn
| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . . . . 6
⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅) |
| 2 | 1 | xkotopon 23608 |
. . . . 5
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆))) |
| 3 | 2 | 3adant3 1133 |
. . . 4
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆))) |
| 4 | | xkopjcn.1 |
. . . . . . . . 9
⊢ 𝑋 = ∪
𝑅 |
| 5 | 4 | topopn 22912 |
. . . . . . . 8
⊢ (𝑅 ∈ Top → 𝑋 ∈ 𝑅) |
| 6 | 5 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝑋 ∈ 𝑅) |
| 7 | | fconst6g 6797 |
. . . . . . . 8
⊢ (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶Top) |
| 8 | 7 | 3ad2ant2 1135 |
. . . . . . 7
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑋 × {𝑆}):𝑋⟶Top) |
| 9 | | pttop 23590 |
. . . . . . 7
⊢ ((𝑋 ∈ 𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) →
(∏t‘(𝑋 × {𝑆})) ∈ Top) |
| 10 | 6, 8, 9 | syl2anc 584 |
. . . . . 6
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∏t‘(𝑋 × {𝑆})) ∈ Top) |
| 11 | | eqid 2737 |
. . . . . . . . . 10
⊢ ∪ 𝑆 =
∪ 𝑆 |
| 12 | 4, 11 | cnf 23254 |
. . . . . . . . 9
⊢ (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓:𝑋⟶∪ 𝑆) |
| 13 | | uniexg 7760 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ Top → ∪ 𝑆
∈ V) |
| 14 | 13 | 3ad2ant2 1135 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ∪ 𝑆 ∈ V) |
| 15 | 14, 6 | elmapd 8880 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (∪ 𝑆 ↑m 𝑋) ↔ 𝑓:𝑋⟶∪ 𝑆)) |
| 16 | 12, 15 | imbitrrid 246 |
. . . . . . . 8
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓 ∈ (∪ 𝑆 ↑m 𝑋))) |
| 17 | 16 | ssrdv 3989 |
. . . . . . 7
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) ⊆ (∪
𝑆 ↑m 𝑋)) |
| 18 | | simp2 1138 |
. . . . . . . 8
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝑆 ∈ Top) |
| 19 | | eqid 2737 |
. . . . . . . . 9
⊢
(∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆})) |
| 20 | 19, 11 | ptuniconst 23606 |
. . . . . . . 8
⊢ ((𝑋 ∈ 𝑅 ∧ 𝑆 ∈ Top) → (∪ 𝑆
↑m 𝑋) =
∪ (∏t‘(𝑋 × {𝑆}))) |
| 21 | 6, 18, 20 | syl2anc 584 |
. . . . . . 7
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∪ 𝑆 ↑m 𝑋) = ∪
(∏t‘(𝑋 × {𝑆}))) |
| 22 | 17, 21 | sseqtrd 4020 |
. . . . . 6
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) ⊆ ∪
(∏t‘(𝑋 × {𝑆}))) |
| 23 | | eqid 2737 |
. . . . . . 7
⊢ ∪ (∏t‘(𝑋 × {𝑆})) = ∪
(∏t‘(𝑋 × {𝑆})) |
| 24 | 23 | restuni 23170 |
. . . . . 6
⊢
(((∏t‘(𝑋 × {𝑆})) ∈ Top ∧ (𝑅 Cn 𝑆) ⊆ ∪
(∏t‘(𝑋 × {𝑆}))) → (𝑅 Cn 𝑆) = ∪
((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))) |
| 25 | 10, 22, 24 | syl2anc 584 |
. . . . 5
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) = ∪
((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))) |
| 26 | 25 | fveq2d 6910 |
. . . 4
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (TopOn‘(𝑅 Cn 𝑆)) = (TopOn‘∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))) |
| 27 | 3, 26 | eleqtrd 2843 |
. . 3
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))) |
| 28 | 4, 19 | xkoptsub 23662 |
. . . 4
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) →
((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅)) |
| 29 | 28 | 3adant3 1133 |
. . 3
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅)) |
| 30 | | eqid 2737 |
. . . 4
⊢ ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) = ∪
((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) |
| 31 | 30 | cnss1 23284 |
. . 3
⊢ (((𝑆 ↑ko 𝑅) ∈ (TopOn‘∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))) ∧ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅)) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆 ↑ko 𝑅) Cn 𝑆)) |
| 32 | 27, 29, 31 | syl2anc 584 |
. 2
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆 ↑ko 𝑅) Cn 𝑆)) |
| 33 | 22 | resmptd 6058 |
. . 3
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑓 ∈ ∪
(∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ↾ (𝑅 Cn 𝑆)) = (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴))) |
| 34 | | simp3 1139 |
. . . . . 6
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) |
| 35 | 23, 19 | ptpjcn 23619 |
. . . . . 6
⊢ ((𝑋 ∈ 𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ ∪
(∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴))) |
| 36 | 6, 8, 34, 35 | syl3anc 1373 |
. . . . 5
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ ∪
(∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴))) |
| 37 | | fvconst2g 7222 |
. . . . . . 7
⊢ ((𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆) |
| 38 | 37 | 3adant1 1131 |
. . . . . 6
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆) |
| 39 | 38 | oveq2d 7447 |
. . . . 5
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)) = ((∏t‘(𝑋 × {𝑆})) Cn 𝑆)) |
| 40 | 36, 39 | eleqtrd 2843 |
. . . 4
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ ∪
(∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆)) |
| 41 | 23 | cnrest 23293 |
. . . 4
⊢ (((𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆) ∧ (𝑅 Cn 𝑆) ⊆ ∪
(∏t‘(𝑋 × {𝑆}))) → ((𝑓 ∈ ∪
(∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ↾ (𝑅 Cn 𝑆)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆)) |
| 42 | 40, 22, 41 | syl2anc 584 |
. . 3
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑓 ∈ ∪
(∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ↾ (𝑅 Cn 𝑆)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆)) |
| 43 | 33, 42 | eqeltrrd 2842 |
. 2
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆)) |
| 44 | 32, 43 | sseldd 3984 |
1
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴)) ∈ ((𝑆 ↑ko 𝑅) Cn 𝑆)) |