| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | filnet.h | . . . . 5
⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) | 
| 2 |  | filnet.d | . . . . 5
⊢ 𝐷 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st
‘𝑥))} | 
| 3 | 1, 2 | filnetlem3 36382 | . . . 4
⊢ (𝐻 = ∪
∪ 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel))) | 
| 4 | 3 | simpri 485 | . . 3
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)) | 
| 5 | 4 | simprd 495 | . 2
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ DirRel) | 
| 6 |  | f2ndres 8040 | . . . . 5
⊢
(2nd ↾ (𝐹 × 𝑋)):(𝐹 × 𝑋)⟶𝑋 | 
| 7 | 4 | simpld 494 | . . . . 5
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐻 ⊆ (𝐹 × 𝑋)) | 
| 8 |  | fssres2 6775 | . . . . 5
⊢
(((2nd ↾ (𝐹 × 𝑋)):(𝐹 × 𝑋)⟶𝑋 ∧ 𝐻 ⊆ (𝐹 × 𝑋)) → (2nd ↾ 𝐻):𝐻⟶𝑋) | 
| 9 | 6, 7, 8 | sylancr 587 | . . . 4
⊢ (𝐹 ∈ (Fil‘𝑋) → (2nd ↾
𝐻):𝐻⟶𝑋) | 
| 10 |  | filtop 23864 | . . . . . 6
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) | 
| 11 |  | xpexg 7771 | . . . . . 6
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑋 ∈ 𝐹) → (𝐹 × 𝑋) ∈ V) | 
| 12 | 10, 11 | mpdan 687 | . . . . 5
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 × 𝑋) ∈ V) | 
| 13 | 12, 7 | ssexd 5323 | . . . 4
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐻 ∈ V) | 
| 14 | 9, 13 | fexd 7248 | . . 3
⊢ (𝐹 ∈ (Fil‘𝑋) → (2nd ↾
𝐻) ∈
V) | 
| 15 | 3 | simpli 483 | . . . . . . 7
⊢ 𝐻 = ∪
∪ 𝐷 | 
| 16 |  | dirdm 18646 | . . . . . . . 8
⊢ (𝐷 ∈ DirRel → dom 𝐷 = ∪
∪ 𝐷) | 
| 17 | 5, 16 | syl 17 | . . . . . . 7
⊢ (𝐹 ∈ (Fil‘𝑋) → dom 𝐷 = ∪ ∪ 𝐷) | 
| 18 | 15, 17 | eqtr4id 2795 | . . . . . 6
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐻 = dom 𝐷) | 
| 19 | 18 | feq2d 6721 | . . . . 5
⊢ (𝐹 ∈ (Fil‘𝑋) → ((2nd
↾ 𝐻):𝐻⟶𝑋 ↔ (2nd ↾ 𝐻):dom 𝐷⟶𝑋)) | 
| 20 | 9, 19 | mpbid 232 | . . . 4
⊢ (𝐹 ∈ (Fil‘𝑋) → (2nd ↾
𝐻):dom 𝐷⟶𝑋) | 
| 21 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢ dom 𝐷 = dom 𝐷 | 
| 22 | 21 | tailf 36377 | . . . . . . . . . . . . 13
⊢ (𝐷 ∈ DirRel →
(tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷) | 
| 23 | 5, 22 | syl 17 | . . . . . . . . . . . 12
⊢ (𝐹 ∈ (Fil‘𝑋) → (tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷) | 
| 24 | 18 | feq2d 6721 | . . . . . . . . . . . 12
⊢ (𝐹 ∈ (Fil‘𝑋) → ((tail‘𝐷):𝐻⟶𝒫 dom 𝐷 ↔ (tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷)) | 
| 25 | 23, 24 | mpbird 257 | . . . . . . . . . . 11
⊢ (𝐹 ∈ (Fil‘𝑋) → (tail‘𝐷):𝐻⟶𝒫 dom 𝐷) | 
| 26 | 25 | adantr 480 | . . . . . . . . . 10
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) → (tail‘𝐷):𝐻⟶𝒫 dom 𝐷) | 
| 27 |  | ffn 6735 | . . . . . . . . . 10
⊢
((tail‘𝐷):𝐻⟶𝒫 dom 𝐷 → (tail‘𝐷) Fn 𝐻) | 
| 28 |  | imaeq2 6073 | . . . . . . . . . . . 12
⊢ (𝑑 = ((tail‘𝐷)‘𝑓) → ((2nd ↾ 𝐻) “ 𝑑) = ((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓))) | 
| 29 | 28 | sseq1d 4014 | . . . . . . . . . . 11
⊢ (𝑑 = ((tail‘𝐷)‘𝑓) → (((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡 ↔ ((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡)) | 
| 30 | 29 | rexrn 7106 | . . . . . . . . . 10
⊢
((tail‘𝐷) Fn
𝐻 → (∃𝑑 ∈ ran (tail‘𝐷)((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡 ↔ ∃𝑓 ∈ 𝐻 ((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡)) | 
| 31 | 26, 27, 30 | 3syl 18 | . . . . . . . . 9
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) → (∃𝑑 ∈ ran (tail‘𝐷)((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡 ↔ ∃𝑓 ∈ 𝐻 ((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡)) | 
| 32 |  | fo2nd 8036 | . . . . . . . . . . . . . . 15
⊢
2nd :V–onto→V | 
| 33 |  | fofn 6821 | . . . . . . . . . . . . . . 15
⊢
(2nd :V–onto→V → 2nd Fn V) | 
| 34 | 32, 33 | ax-mp 5 | . . . . . . . . . . . . . 14
⊢
2nd Fn V | 
| 35 |  | ssv 4007 | . . . . . . . . . . . . . 14
⊢ 𝐻 ⊆ V | 
| 36 |  | fnssres 6690 | . . . . . . . . . . . . . 14
⊢
((2nd Fn V ∧ 𝐻 ⊆ V) → (2nd ↾
𝐻) Fn 𝐻) | 
| 37 | 34, 35, 36 | mp2an 692 | . . . . . . . . . . . . 13
⊢
(2nd ↾ 𝐻) Fn 𝐻 | 
| 38 |  | fnfun 6667 | . . . . . . . . . . . . 13
⊢
((2nd ↾ 𝐻) Fn 𝐻 → Fun (2nd ↾ 𝐻)) | 
| 39 | 37, 38 | ax-mp 5 | . . . . . . . . . . . 12
⊢ Fun
(2nd ↾ 𝐻) | 
| 40 | 26 | ffvelcdmda 7103 | . . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((tail‘𝐷)‘𝑓) ∈ 𝒫 dom 𝐷) | 
| 41 | 40 | elpwid 4608 | . . . . . . . . . . . . . 14
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((tail‘𝐷)‘𝑓) ⊆ dom 𝐷) | 
| 42 | 18 | ad2antrr 726 | . . . . . . . . . . . . . 14
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → 𝐻 = dom 𝐷) | 
| 43 | 41, 42 | sseqtrrd 4020 | . . . . . . . . . . . . 13
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((tail‘𝐷)‘𝑓) ⊆ 𝐻) | 
| 44 | 37 | fndmi 6671 | . . . . . . . . . . . . 13
⊢ dom
(2nd ↾ 𝐻)
= 𝐻 | 
| 45 | 43, 44 | sseqtrrdi 4024 | . . . . . . . . . . . 12
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((tail‘𝐷)‘𝑓) ⊆ dom (2nd ↾ 𝐻)) | 
| 46 |  | funimass4 6972 | . . . . . . . . . . . 12
⊢ ((Fun
(2nd ↾ 𝐻)
∧ ((tail‘𝐷)‘𝑓) ⊆ dom (2nd ↾ 𝐻)) → (((2nd
↾ 𝐻) “
((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡)) | 
| 47 | 39, 45, 46 | sylancr 587 | . . . . . . . . . . 11
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → (((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡)) | 
| 48 | 5 | ad2antrr 726 | . . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → 𝐷 ∈ DirRel) | 
| 49 |  | simpr 484 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → 𝑓 ∈ 𝐻) | 
| 50 | 49, 42 | eleqtrd 2842 | . . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → 𝑓 ∈ dom 𝐷) | 
| 51 |  | vex 3483 | . . . . . . . . . . . . . . . . 17
⊢ 𝑑 ∈ V | 
| 52 | 51 | a1i 11 | . . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → 𝑑 ∈ V) | 
| 53 | 21 | eltail 36376 | . . . . . . . . . . . . . . . 16
⊢ ((𝐷 ∈ DirRel ∧ 𝑓 ∈ dom 𝐷 ∧ 𝑑 ∈ V) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ 𝑓𝐷𝑑)) | 
| 54 | 48, 50, 52, 53 | syl3anc 1372 | . . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ 𝑓𝐷𝑑)) | 
| 55 | 49 | biantrurd 532 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → (𝑑 ∈ 𝐻 ↔ (𝑓 ∈ 𝐻 ∧ 𝑑 ∈ 𝐻))) | 
| 56 | 55 | anbi1d 631 | . . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st
‘𝑓)) ↔ ((𝑓 ∈ 𝐻 ∧ 𝑑 ∈ 𝐻) ∧ (1st ‘𝑑) ⊆ (1st
‘𝑓)))) | 
| 57 |  | vex 3483 | . . . . . . . . . . . . . . . . 17
⊢ 𝑓 ∈ V | 
| 58 | 1, 2, 57, 51 | filnetlem1 36380 | . . . . . . . . . . . . . . . 16
⊢ (𝑓𝐷𝑑 ↔ ((𝑓 ∈ 𝐻 ∧ 𝑑 ∈ 𝐻) ∧ (1st ‘𝑑) ⊆ (1st
‘𝑓))) | 
| 59 | 56, 58 | bitr4di 289 | . . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st
‘𝑓)) ↔ 𝑓𝐷𝑑)) | 
| 60 | 54, 59 | bitr4d 282 | . . . . . . . . . . . . . 14
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ (𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st
‘𝑓)))) | 
| 61 | 60 | imbi1d 341 | . . . . . . . . . . . . 13
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((𝑑 ∈ ((tail‘𝐷)‘𝑓) → ((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡) ↔ ((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st
‘𝑓)) →
((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡))) | 
| 62 |  | fvres 6924 | . . . . . . . . . . . . . . . . 17
⊢ (𝑑 ∈ 𝐻 → ((2nd ↾ 𝐻)‘𝑑) = (2nd ‘𝑑)) | 
| 63 | 62 | eleq1d 2825 | . . . . . . . . . . . . . . . 16
⊢ (𝑑 ∈ 𝐻 → (((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡 ↔ (2nd ‘𝑑) ∈ 𝑡)) | 
| 64 | 63 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st
‘𝑓)) →
(((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡 ↔ (2nd ‘𝑑) ∈ 𝑡)) | 
| 65 | 64 | pm5.74i 271 | . . . . . . . . . . . . . 14
⊢ (((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st
‘𝑓)) →
((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡) ↔ ((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st
‘𝑓)) →
(2nd ‘𝑑)
∈ 𝑡)) | 
| 66 |  | impexp 450 | . . . . . . . . . . . . . 14
⊢ (((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st
‘𝑓)) →
(2nd ‘𝑑)
∈ 𝑡) ↔ (𝑑 ∈ 𝐻 → ((1st ‘𝑑) ⊆ (1st
‘𝑓) →
(2nd ‘𝑑)
∈ 𝑡))) | 
| 67 | 65, 66 | bitri 275 | . . . . . . . . . . . . 13
⊢ (((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st
‘𝑓)) →
((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡) ↔ (𝑑 ∈ 𝐻 → ((1st ‘𝑑) ⊆ (1st
‘𝑓) →
(2nd ‘𝑑)
∈ 𝑡))) | 
| 68 | 61, 67 | bitrdi 287 | . . . . . . . . . . . 12
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((𝑑 ∈ ((tail‘𝐷)‘𝑓) → ((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡) ↔ (𝑑 ∈ 𝐻 → ((1st ‘𝑑) ⊆ (1st
‘𝑓) →
(2nd ‘𝑑)
∈ 𝑡)))) | 
| 69 | 68 | ralbidv2 3173 | . . . . . . . . . . 11
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → (∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡 ↔ ∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd
‘𝑑) ∈ 𝑡))) | 
| 70 | 47, 69 | bitrd 279 | . . . . . . . . . 10
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → (((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd
‘𝑑) ∈ 𝑡))) | 
| 71 | 70 | rexbidva 3176 | . . . . . . . . 9
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) → (∃𝑓 ∈ 𝐻 ((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∃𝑓 ∈ 𝐻 ∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd
‘𝑑) ∈ 𝑡))) | 
| 72 |  | vex 3483 | . . . . . . . . . . . . . . . . 17
⊢ 𝑘 ∈ V | 
| 73 |  | vex 3483 | . . . . . . . . . . . . . . . . 17
⊢ 𝑣 ∈ V | 
| 74 | 72, 73 | op1std 8025 | . . . . . . . . . . . . . . . 16
⊢ (𝑑 = 〈𝑘, 𝑣〉 → (1st ‘𝑑) = 𝑘) | 
| 75 | 74 | sseq1d 4014 | . . . . . . . . . . . . . . 15
⊢ (𝑑 = 〈𝑘, 𝑣〉 → ((1st ‘𝑑) ⊆ (1st
‘𝑓) ↔ 𝑘 ⊆ (1st
‘𝑓))) | 
| 76 | 72, 73 | op2ndd 8026 | . . . . . . . . . . . . . . . 16
⊢ (𝑑 = 〈𝑘, 𝑣〉 → (2nd ‘𝑑) = 𝑣) | 
| 77 | 76 | eleq1d 2825 | . . . . . . . . . . . . . . 15
⊢ (𝑑 = 〈𝑘, 𝑣〉 → ((2nd ‘𝑑) ∈ 𝑡 ↔ 𝑣 ∈ 𝑡)) | 
| 78 | 75, 77 | imbi12d 344 | . . . . . . . . . . . . . 14
⊢ (𝑑 = 〈𝑘, 𝑣〉 → (((1st ‘𝑑) ⊆ (1st
‘𝑓) →
(2nd ‘𝑑)
∈ 𝑡) ↔ (𝑘 ⊆ (1st
‘𝑓) → 𝑣 ∈ 𝑡))) | 
| 79 | 78 | raliunxp 5849 | . . . . . . . . . . . . 13
⊢
(∀𝑑 ∈
∪ 𝑘 ∈ 𝐹 ({𝑘} × 𝑘)((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd
‘𝑑) ∈ 𝑡) ↔ ∀𝑘 ∈ 𝐹 ∀𝑣 ∈ 𝑘 (𝑘 ⊆ (1st ‘𝑓) → 𝑣 ∈ 𝑡)) | 
| 80 |  | sneq 4635 | . . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑘 → {𝑛} = {𝑘}) | 
| 81 |  | id 22 | . . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘) | 
| 82 | 80, 81 | xpeq12d 5715 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → ({𝑛} × 𝑛) = ({𝑘} × 𝑘)) | 
| 83 | 82 | cbviunv 5039 | . . . . . . . . . . . . . . 15
⊢ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) = ∪ 𝑘 ∈ 𝐹 ({𝑘} × 𝑘) | 
| 84 | 1, 83 | eqtri 2764 | . . . . . . . . . . . . . 14
⊢ 𝐻 = ∪ 𝑘 ∈ 𝐹 ({𝑘} × 𝑘) | 
| 85 | 84 | raleqi 3323 | . . . . . . . . . . . . 13
⊢
(∀𝑑 ∈
𝐻 ((1st
‘𝑑) ⊆
(1st ‘𝑓)
→ (2nd ‘𝑑) ∈ 𝑡) ↔ ∀𝑑 ∈ ∪
𝑘 ∈ 𝐹 ({𝑘} × 𝑘)((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd
‘𝑑) ∈ 𝑡)) | 
| 86 |  | dfss3 3971 | . . . . . . . . . . . . . . . 16
⊢ (𝑘 ⊆ 𝑡 ↔ ∀𝑣 ∈ 𝑘 𝑣 ∈ 𝑡) | 
| 87 | 86 | imbi2i 336 | . . . . . . . . . . . . . . 15
⊢ ((𝑘 ⊆ (1st
‘𝑓) → 𝑘 ⊆ 𝑡) ↔ (𝑘 ⊆ (1st ‘𝑓) → ∀𝑣 ∈ 𝑘 𝑣 ∈ 𝑡)) | 
| 88 |  | r19.21v 3179 | . . . . . . . . . . . . . . 15
⊢
(∀𝑣 ∈
𝑘 (𝑘 ⊆ (1st ‘𝑓) → 𝑣 ∈ 𝑡) ↔ (𝑘 ⊆ (1st ‘𝑓) → ∀𝑣 ∈ 𝑘 𝑣 ∈ 𝑡)) | 
| 89 | 87, 88 | bitr4i 278 | . . . . . . . . . . . . . 14
⊢ ((𝑘 ⊆ (1st
‘𝑓) → 𝑘 ⊆ 𝑡) ↔ ∀𝑣 ∈ 𝑘 (𝑘 ⊆ (1st ‘𝑓) → 𝑣 ∈ 𝑡)) | 
| 90 | 89 | ralbii 3092 | . . . . . . . . . . . . 13
⊢
(∀𝑘 ∈
𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡) ↔ ∀𝑘 ∈ 𝐹 ∀𝑣 ∈ 𝑘 (𝑘 ⊆ (1st ‘𝑓) → 𝑣 ∈ 𝑡)) | 
| 91 | 79, 85, 90 | 3bitr4i 303 | . . . . . . . . . . . 12
⊢
(∀𝑑 ∈
𝐻 ((1st
‘𝑑) ⊆
(1st ‘𝑓)
→ (2nd ‘𝑑) ∈ 𝑡) ↔ ∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡)) | 
| 92 | 91 | rexbii 3093 | . . . . . . . . . . 11
⊢
(∃𝑓 ∈
𝐻 ∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd
‘𝑑) ∈ 𝑡) ↔ ∃𝑓 ∈ 𝐻 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡)) | 
| 93 | 1 | rexeqi 3324 | . . . . . . . . . . 11
⊢
(∃𝑓 ∈
𝐻 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡) ↔ ∃𝑓 ∈ ∪
𝑛 ∈ 𝐹 ({𝑛} × 𝑛)∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡)) | 
| 94 |  | vex 3483 | . . . . . . . . . . . . . . . 16
⊢ 𝑛 ∈ V | 
| 95 |  | vex 3483 | . . . . . . . . . . . . . . . 16
⊢ 𝑚 ∈ V | 
| 96 | 94, 95 | op1std 8025 | . . . . . . . . . . . . . . 15
⊢ (𝑓 = 〈𝑛, 𝑚〉 → (1st ‘𝑓) = 𝑛) | 
| 97 | 96 | sseq2d 4015 | . . . . . . . . . . . . . 14
⊢ (𝑓 = 〈𝑛, 𝑚〉 → (𝑘 ⊆ (1st ‘𝑓) ↔ 𝑘 ⊆ 𝑛)) | 
| 98 | 97 | imbi1d 341 | . . . . . . . . . . . . 13
⊢ (𝑓 = 〈𝑛, 𝑚〉 → ((𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡) ↔ (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡))) | 
| 99 | 98 | ralbidv 3177 | . . . . . . . . . . . 12
⊢ (𝑓 = 〈𝑛, 𝑚〉 → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡) ↔ ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡))) | 
| 100 | 99 | rexiunxp 5850 | . . . . . . . . . . 11
⊢
(∃𝑓 ∈
∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡) ↔ ∃𝑛 ∈ 𝐹 ∃𝑚 ∈ 𝑛 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡)) | 
| 101 | 92, 93, 100 | 3bitri 297 | . . . . . . . . . 10
⊢
(∃𝑓 ∈
𝐻 ∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd
‘𝑑) ∈ 𝑡) ↔ ∃𝑛 ∈ 𝐹 ∃𝑚 ∈ 𝑛 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡)) | 
| 102 |  | fileln0 23859 | . . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ 𝐹) → 𝑛 ≠ ∅) | 
| 103 | 102 | adantlr 715 | . . . . . . . . . . . . 13
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑛 ∈ 𝐹) → 𝑛 ≠ ∅) | 
| 104 |  | r19.9rzv 4499 | . . . . . . . . . . . . 13
⊢ (𝑛 ≠ ∅ →
(∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) ↔ ∃𝑚 ∈ 𝑛 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡))) | 
| 105 | 103, 104 | syl 17 | . . . . . . . . . . . 12
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑛 ∈ 𝐹) → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) ↔ ∃𝑚 ∈ 𝑛 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡))) | 
| 106 |  | ssid 4005 | . . . . . . . . . . . . . . 15
⊢ 𝑛 ⊆ 𝑛 | 
| 107 |  | sseq1 4008 | . . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → (𝑘 ⊆ 𝑛 ↔ 𝑛 ⊆ 𝑛)) | 
| 108 |  | sseq1 4008 | . . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → (𝑘 ⊆ 𝑡 ↔ 𝑛 ⊆ 𝑡)) | 
| 109 | 107, 108 | imbi12d 344 | . . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → ((𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) ↔ (𝑛 ⊆ 𝑛 → 𝑛 ⊆ 𝑡))) | 
| 110 | 109 | rspcv 3617 | . . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ 𝐹 → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) → (𝑛 ⊆ 𝑛 → 𝑛 ⊆ 𝑡))) | 
| 111 | 106, 110 | mpii 46 | . . . . . . . . . . . . . 14
⊢ (𝑛 ∈ 𝐹 → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) → 𝑛 ⊆ 𝑡)) | 
| 112 | 111 | adantl 481 | . . . . . . . . . . . . 13
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑛 ∈ 𝐹) → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) → 𝑛 ⊆ 𝑡)) | 
| 113 |  | sstr2 3989 | . . . . . . . . . . . . . . 15
⊢ (𝑘 ⊆ 𝑛 → (𝑛 ⊆ 𝑡 → 𝑘 ⊆ 𝑡)) | 
| 114 | 113 | com12 32 | . . . . . . . . . . . . . 14
⊢ (𝑛 ⊆ 𝑡 → (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡)) | 
| 115 | 114 | ralrimivw 3149 | . . . . . . . . . . . . 13
⊢ (𝑛 ⊆ 𝑡 → ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡)) | 
| 116 | 112, 115 | impbid1 225 | . . . . . . . . . . . 12
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑛 ∈ 𝐹) → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) ↔ 𝑛 ⊆ 𝑡)) | 
| 117 | 105, 116 | bitr3d 281 | . . . . . . . . . . 11
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑛 ∈ 𝐹) → (∃𝑚 ∈ 𝑛 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) ↔ 𝑛 ⊆ 𝑡)) | 
| 118 | 117 | rexbidva 3176 | . . . . . . . . . 10
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) → (∃𝑛 ∈ 𝐹 ∃𝑚 ∈ 𝑛 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) ↔ ∃𝑛 ∈ 𝐹 𝑛 ⊆ 𝑡)) | 
| 119 | 101, 118 | bitrid 283 | . . . . . . . . 9
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) → (∃𝑓 ∈ 𝐻 ∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd
‘𝑑) ∈ 𝑡) ↔ ∃𝑛 ∈ 𝐹 𝑛 ⊆ 𝑡)) | 
| 120 | 31, 71, 119 | 3bitrd 305 | . . . . . . . 8
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) → (∃𝑑 ∈ ran (tail‘𝐷)((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡 ↔ ∃𝑛 ∈ 𝐹 𝑛 ⊆ 𝑡)) | 
| 121 | 120 | pm5.32da 579 | . . . . . . 7
⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑡 ⊆ 𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑛 ∈ 𝐹 𝑛 ⊆ 𝑡))) | 
| 122 |  | filn0 23871 | . . . . . . . . . . . 12
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅) | 
| 123 | 94 | snnz 4775 | . . . . . . . . . . . . . . . 16
⊢ {𝑛} ≠ ∅ | 
| 124 | 102, 123 | jctil 519 | . . . . . . . . . . . . . . 15
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ 𝐹) → ({𝑛} ≠ ∅ ∧ 𝑛 ≠ ∅)) | 
| 125 |  | neanior 3034 | . . . . . . . . . . . . . . 15
⊢ (({𝑛} ≠ ∅ ∧ 𝑛 ≠ ∅) ↔ ¬
({𝑛} = ∅ ∨ 𝑛 = ∅)) | 
| 126 | 124, 125 | sylib 218 | . . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ 𝐹) → ¬ ({𝑛} = ∅ ∨ 𝑛 = ∅)) | 
| 127 |  | ss0b 4400 | . . . . . . . . . . . . . . 15
⊢ (({𝑛} × 𝑛) ⊆ ∅ ↔ ({𝑛} × 𝑛) = ∅) | 
| 128 |  | xpeq0 6179 | . . . . . . . . . . . . . . 15
⊢ (({𝑛} × 𝑛) = ∅ ↔ ({𝑛} = ∅ ∨ 𝑛 = ∅)) | 
| 129 | 127, 128 | bitri 275 | . . . . . . . . . . . . . 14
⊢ (({𝑛} × 𝑛) ⊆ ∅ ↔ ({𝑛} = ∅ ∨ 𝑛 = ∅)) | 
| 130 | 126, 129 | sylnibr 329 | . . . . . . . . . . . . 13
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ 𝐹) → ¬ ({𝑛} × 𝑛) ⊆ ∅) | 
| 131 | 130 | ralrimiva 3145 | . . . . . . . . . . . 12
⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑛 ∈ 𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅) | 
| 132 |  | r19.2z 4494 | . . . . . . . . . . . 12
⊢ ((𝐹 ≠ ∅ ∧
∀𝑛 ∈ 𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅) → ∃𝑛 ∈ 𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅) | 
| 133 | 122, 131,
132 | syl2anc 584 | . . . . . . . . . . 11
⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑛 ∈ 𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅) | 
| 134 |  | rexnal 3099 | . . . . . . . . . . 11
⊢
(∃𝑛 ∈
𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅ ↔ ¬ ∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅) | 
| 135 | 133, 134 | sylib 218 | . . . . . . . . . 10
⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅) | 
| 136 | 1 | sseq1i 4011 | . . . . . . . . . . . 12
⊢ (𝐻 ⊆ ∅ ↔ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅) | 
| 137 |  | ss0b 4400 | . . . . . . . . . . . 12
⊢ (𝐻 ⊆ ∅ ↔ 𝐻 = ∅) | 
| 138 |  | iunss 5044 | . . . . . . . . . . . 12
⊢ (∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅ ↔ ∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅) | 
| 139 | 136, 137,
138 | 3bitr3i 301 | . . . . . . . . . . 11
⊢ (𝐻 = ∅ ↔ ∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅) | 
| 140 | 139 | necon3abii 2986 | . . . . . . . . . 10
⊢ (𝐻 ≠ ∅ ↔ ¬
∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅) | 
| 141 | 135, 140 | sylibr 234 | . . . . . . . . 9
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐻 ≠ ∅) | 
| 142 |  | dmresi 6069 | . . . . . . . . . . . 12
⊢ dom ( I
↾ 𝐻) = 𝐻 | 
| 143 | 1, 2 | filnetlem2 36381 | . . . . . . . . . . . . . 14
⊢ (( I
↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻)) | 
| 144 | 143 | simpli 483 | . . . . . . . . . . . . 13
⊢ ( I
↾ 𝐻) ⊆ 𝐷 | 
| 145 |  | dmss 5912 | . . . . . . . . . . . . 13
⊢ (( I
↾ 𝐻) ⊆ 𝐷 → dom ( I ↾ 𝐻) ⊆ dom 𝐷) | 
| 146 | 144, 145 | ax-mp 5 | . . . . . . . . . . . 12
⊢ dom ( I
↾ 𝐻) ⊆ dom
𝐷 | 
| 147 | 142, 146 | eqsstrri 4030 | . . . . . . . . . . 11
⊢ 𝐻 ⊆ dom 𝐷 | 
| 148 | 143 | simpri 485 | . . . . . . . . . . . . 13
⊢ 𝐷 ⊆ (𝐻 × 𝐻) | 
| 149 |  | dmss 5912 | . . . . . . . . . . . . 13
⊢ (𝐷 ⊆ (𝐻 × 𝐻) → dom 𝐷 ⊆ dom (𝐻 × 𝐻)) | 
| 150 | 148, 149 | ax-mp 5 | . . . . . . . . . . . 12
⊢ dom 𝐷 ⊆ dom (𝐻 × 𝐻) | 
| 151 |  | dmxpid 5940 | . . . . . . . . . . . 12
⊢ dom
(𝐻 × 𝐻) = 𝐻 | 
| 152 | 150, 151 | sseqtri 4031 | . . . . . . . . . . 11
⊢ dom 𝐷 ⊆ 𝐻 | 
| 153 | 147, 152 | eqssi 3999 | . . . . . . . . . 10
⊢ 𝐻 = dom 𝐷 | 
| 154 | 153 | tailfb 36379 | . . . . . . . . 9
⊢ ((𝐷 ∈ DirRel ∧ 𝐻 ≠ ∅) → ran
(tail‘𝐷) ∈
(fBas‘𝐻)) | 
| 155 | 5, 141, 154 | syl2anc 584 | . . . . . . . 8
⊢ (𝐹 ∈ (Fil‘𝑋) → ran (tail‘𝐷) ∈ (fBas‘𝐻)) | 
| 156 |  | elfm 23956 | . . . . . . . 8
⊢ ((𝑋 ∈ 𝐹 ∧ ran (tail‘𝐷) ∈ (fBas‘𝐻) ∧ (2nd ↾ 𝐻):𝐻⟶𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡))) | 
| 157 | 10, 155, 9, 156 | syl3anc 1372 | . . . . . . 7
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡))) | 
| 158 |  | filfbas 23857 | . . . . . . . 8
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | 
| 159 |  | elfg 23880 | . . . . . . . 8
⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑛 ∈ 𝐹 𝑛 ⊆ 𝑡))) | 
| 160 | 158, 159 | syl 17 | . . . . . . 7
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑛 ∈ 𝐹 𝑛 ⊆ 𝑡))) | 
| 161 | 121, 157,
160 | 3bitr4d 311 | . . . . . 6
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)) ↔ 𝑡 ∈ (𝑋filGen𝐹))) | 
| 162 | 161 | eqrdv 2734 | . . . . 5
⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)) = (𝑋filGen𝐹)) | 
| 163 |  | fgfil 23884 | . . . . 5
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹) | 
| 164 | 162, 163 | eqtr2d 2777 | . . . 4
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 = ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷))) | 
| 165 | 20, 164 | jca 511 | . . 3
⊢ (𝐹 ∈ (Fil‘𝑋) → ((2nd
↾ 𝐻):dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)))) | 
| 166 |  | feq1 6715 | . . . . 5
⊢ (𝑓 = (2nd ↾ 𝐻) → (𝑓:dom 𝐷⟶𝑋 ↔ (2nd ↾ 𝐻):dom 𝐷⟶𝑋)) | 
| 167 |  | oveq2 7440 | . . . . . . 7
⊢ (𝑓 = (2nd ↾ 𝐻) → (𝑋 FilMap 𝑓) = (𝑋 FilMap (2nd ↾ 𝐻))) | 
| 168 | 167 | fveq1d 6907 | . . . . . 6
⊢ (𝑓 = (2nd ↾ 𝐻) → ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)) = ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷))) | 
| 169 | 168 | eqeq2d 2747 | . . . . 5
⊢ (𝑓 = (2nd ↾ 𝐻) → (𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)) ↔ 𝐹 = ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)))) | 
| 170 | 166, 169 | anbi12d 632 | . . . 4
⊢ (𝑓 = (2nd ↾ 𝐻) → ((𝑓:dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))) ↔ ((2nd ↾ 𝐻):dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷))))) | 
| 171 | 170 | spcegv 3596 | . . 3
⊢
((2nd ↾ 𝐻) ∈ V → (((2nd ↾
𝐻):dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷))) → ∃𝑓(𝑓:dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))))) | 
| 172 | 14, 165, 171 | sylc 65 | . 2
⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑓(𝑓:dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))) | 
| 173 |  | dmeq 5913 | . . . . . 6
⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷) | 
| 174 | 173 | feq2d 6721 | . . . . 5
⊢ (𝑑 = 𝐷 → (𝑓:dom 𝑑⟶𝑋 ↔ 𝑓:dom 𝐷⟶𝑋)) | 
| 175 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑑 = 𝐷 → (tail‘𝑑) = (tail‘𝐷)) | 
| 176 | 175 | rneqd 5948 | . . . . . . 7
⊢ (𝑑 = 𝐷 → ran (tail‘𝑑) = ran (tail‘𝐷)) | 
| 177 | 176 | fveq2d 6909 | . . . . . 6
⊢ (𝑑 = 𝐷 → ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)) = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))) | 
| 178 | 177 | eqeq2d 2747 | . . . . 5
⊢ (𝑑 = 𝐷 → (𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)) ↔ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))) | 
| 179 | 174, 178 | anbi12d 632 | . . . 4
⊢ (𝑑 = 𝐷 → ((𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))) ↔ (𝑓:dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))))) | 
| 180 | 179 | exbidv 1920 | . . 3
⊢ (𝑑 = 𝐷 → (∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))) ↔ ∃𝑓(𝑓:dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))))) | 
| 181 | 180 | rspcev 3621 | . 2
⊢ ((𝐷 ∈ DirRel ∧
∃𝑓(𝑓:dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)))) | 
| 182 | 5, 172, 181 | syl2anc 584 | 1
⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)))) |