| Step | Hyp | Ref
| Expression |
| 1 | | ptcld.c |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘(𝐹‘𝑘))) |
| 2 | | eqid 2737 |
. . . . . 6
⊢ ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑘) |
| 3 | 2 | cldss 23037 |
. . . . 5
⊢ (𝐶 ∈ (Clsd‘(𝐹‘𝑘)) → 𝐶 ⊆ ∪ (𝐹‘𝑘)) |
| 4 | 1, 3 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ⊆ ∪ (𝐹‘𝑘)) |
| 5 | 4 | ralrimiva 3146 |
. . 3
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘)) |
| 6 | | boxriin 8980 |
. . 3
⊢
(∀𝑘 ∈
𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘) → X𝑘 ∈ 𝐴 𝐶 = (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∩ ∩
𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)))) |
| 7 | 5, 6 | syl 17 |
. 2
⊢ (𝜑 → X𝑘 ∈
𝐴 𝐶 = (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∩ ∩
𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)))) |
| 8 | | ptcld.a |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| 9 | | ptcld.f |
. . . . 5
⊢ (𝜑 → 𝐹:𝐴⟶Top) |
| 10 | | eqid 2737 |
. . . . . 6
⊢
(∏t‘𝐹) = (∏t‘𝐹) |
| 11 | 10 | ptuni 23602 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈
𝐴 ∪ (𝐹‘𝑘) = ∪
(∏t‘𝐹)) |
| 12 | 8, 9, 11 | syl2anc 584 |
. . . 4
⊢ (𝜑 → X𝑘 ∈
𝐴 ∪ (𝐹‘𝑘) = ∪
(∏t‘𝐹)) |
| 13 | 12 | ineq1d 4219 |
. . 3
⊢ (𝜑 → (X𝑘 ∈
𝐴 ∪ (𝐹‘𝑘) ∩ ∩
𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) = (∪
(∏t‘𝐹) ∩ ∩
𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)))) |
| 14 | | pttop 23590 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) →
(∏t‘𝐹) ∈ Top) |
| 15 | 8, 9, 14 | syl2anc 584 |
. . . 4
⊢ (𝜑 →
(∏t‘𝐹) ∈ Top) |
| 16 | | sseq1 4009 |
. . . . . . . . . . 11
⊢ (𝐶 = if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) → (𝐶 ⊆ ∪ (𝐹‘𝑘) ↔ if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪
(𝐹‘𝑘))) |
| 17 | | sseq1 4009 |
. . . . . . . . . . 11
⊢ (∪ (𝐹‘𝑘) = if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) → (∪
(𝐹‘𝑘) ⊆ ∪ (𝐹‘𝑘) ↔ if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪
(𝐹‘𝑘))) |
| 18 | | simpl 482 |
. . . . . . . . . . 11
⊢ ((𝐶 ⊆ ∪ (𝐹‘𝑘) ∧ 𝑘 = 𝑥) → 𝐶 ⊆ ∪ (𝐹‘𝑘)) |
| 19 | | ssidd 4007 |
. . . . . . . . . . 11
⊢ ((𝐶 ⊆ ∪ (𝐹‘𝑘) ∧ ¬ 𝑘 = 𝑥) → ∪ (𝐹‘𝑘) ⊆ ∪ (𝐹‘𝑘)) |
| 20 | 16, 17, 18, 19 | ifbothda 4564 |
. . . . . . . . . 10
⊢ (𝐶 ⊆ ∪ (𝐹‘𝑘) → if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪
(𝐹‘𝑘)) |
| 21 | 20 | ralimi 3083 |
. . . . . . . . 9
⊢
(∀𝑘 ∈
𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘) → ∀𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪
(𝐹‘𝑘)) |
| 22 | | ss2ixp 8950 |
. . . . . . . . 9
⊢
(∀𝑘 ∈
𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪
(𝐹‘𝑘) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) |
| 23 | 5, 21, 22 | 3syl 18 |
. . . . . . . 8
⊢ (𝜑 → X𝑘 ∈
𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) |
| 24 | 23 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) |
| 25 | 12 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪
(∏t‘𝐹)) |
| 26 | 24, 25 | sseqtrd 4020 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪
(∏t‘𝐹)) |
| 27 | 12 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ (∏t‘𝐹) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) |
| 28 | 27 | difeq1d 4125 |
. . . . . . . . 9
⊢ (𝜑 → (∪ (∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) = (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)))) |
| 29 | 28 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪
(∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) = (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)))) |
| 30 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) |
| 31 | 5 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘)) |
| 32 | | boxcutc 8981 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘)) → (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘))) |
| 33 | 30, 31, 32 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘))) |
| 34 | | ixpeq2 8951 |
. . . . . . . . . 10
⊢
(∀𝑘 ∈
𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘)) = if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘)) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘)) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘))) |
| 35 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥)) |
| 36 | 35 | unieqd 4920 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑥 → ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑥)) |
| 37 | | csbeq1a 3913 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑥 → 𝐶 = ⦋𝑥 / 𝑘⦌𝐶) |
| 38 | 36, 37 | difeq12d 4127 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑥 → (∪ (𝐹‘𝑘) ∖ 𝐶) = (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶)) |
| 39 | 38 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ 𝐴 ∧ 𝑘 = 𝑥) → (∪ (𝐹‘𝑘) ∖ 𝐶) = (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶)) |
| 40 | 39 | ifeq1da 4557 |
. . . . . . . . . 10
⊢ (𝑘 ∈ 𝐴 → if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘)) = if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘))) |
| 41 | 34, 40 | mprg 3067 |
. . . . . . . . 9
⊢ X𝑘 ∈
𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘)) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘)) |
| 42 | 41 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘)) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘))) |
| 43 | 29, 33, 42 | 3eqtrd 2781 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪
(∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘))) |
| 44 | 8 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ 𝑉) |
| 45 | 9 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶Top) |
| 46 | 1 | ralrimiva 3146 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(𝐹‘𝑘))) |
| 47 | | nfv 1914 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥 𝐶 ∈ (Clsd‘(𝐹‘𝑘)) |
| 48 | | nfcsb1v 3923 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶 |
| 49 | 48 | nfel1 2922 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥)) |
| 50 | | 2fveq3 6911 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑥 → (Clsd‘(𝐹‘𝑘)) = (Clsd‘(𝐹‘𝑥))) |
| 51 | 37, 50 | eleq12d 2835 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑥 → (𝐶 ∈ (Clsd‘(𝐹‘𝑘)) ↔ ⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥)))) |
| 52 | 47, 49, 51 | cbvralw 3306 |
. . . . . . . . . . 11
⊢
(∀𝑘 ∈
𝐴 𝐶 ∈ (Clsd‘(𝐹‘𝑘)) ↔ ∀𝑥 ∈ 𝐴 ⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥))) |
| 53 | 46, 52 | sylib 218 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥))) |
| 54 | 53 | r19.21bi 3251 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥))) |
| 55 | | eqid 2737 |
. . . . . . . . . 10
⊢ ∪ (𝐹‘𝑥) = ∪ (𝐹‘𝑥) |
| 56 | 55 | cldopn 23039 |
. . . . . . . . 9
⊢
(⦋𝑥 /
𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥)) → (∪
(𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶) ∈ (𝐹‘𝑥)) |
| 57 | 54, 56 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪
(𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶) ∈ (𝐹‘𝑥)) |
| 58 | 44, 45, 57 | ptopn2 23592 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘)) ∈ (∏t‘𝐹)) |
| 59 | 43, 58 | eqeltrd 2841 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪
(∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (∏t‘𝐹)) |
| 60 | | eqid 2737 |
. . . . . . . . 9
⊢ ∪ (∏t‘𝐹) = ∪
(∏t‘𝐹) |
| 61 | 60 | iscld 23035 |
. . . . . . . 8
⊢
((∏t‘𝐹) ∈ Top → (X𝑘 ∈
𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈
(Clsd‘(∏t‘𝐹)) ↔ (X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪
(∏t‘𝐹) ∧ (∪
(∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (∏t‘𝐹)))) |
| 62 | 15, 61 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (X𝑘 ∈
𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈
(Clsd‘(∏t‘𝐹)) ↔ (X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪
(∏t‘𝐹) ∧ (∪
(∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (∏t‘𝐹)))) |
| 63 | 62 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈
(Clsd‘(∏t‘𝐹)) ↔ (X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪
(∏t‘𝐹) ∧ (∪
(∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (∏t‘𝐹)))) |
| 64 | 26, 59, 63 | mpbir2and 713 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈
(Clsd‘(∏t‘𝐹))) |
| 65 | 64 | ralrimiva 3146 |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈
(Clsd‘(∏t‘𝐹))) |
| 66 | 60 | riincld 23052 |
. . . 4
⊢
(((∏t‘𝐹) ∈ Top ∧ ∀𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈
(Clsd‘(∏t‘𝐹))) → (∪
(∏t‘𝐹) ∩ ∩
𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈
(Clsd‘(∏t‘𝐹))) |
| 67 | 15, 65, 66 | syl2anc 584 |
. . 3
⊢ (𝜑 → (∪ (∏t‘𝐹) ∩ ∩
𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈
(Clsd‘(∏t‘𝐹))) |
| 68 | 13, 67 | eqeltrd 2841 |
. 2
⊢ (𝜑 → (X𝑘 ∈
𝐴 ∪ (𝐹‘𝑘) ∩ ∩
𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈
(Clsd‘(∏t‘𝐹))) |
| 69 | 7, 68 | eqeltrd 2841 |
1
⊢ (𝜑 → X𝑘 ∈
𝐴 𝐶 ∈
(Clsd‘(∏t‘𝐹))) |