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Theorem filnetlem4 36364
Description: Lemma for filnet 36365. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
Hypotheses
Ref Expression
filnet.h 𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)
filnet.d 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
Assertion
Ref Expression
filnetlem4 (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))
Distinct variable groups:   𝑥,𝑦   𝑓,𝑑,𝑛,𝑥,𝑦,𝐹   𝐻,𝑑,𝑓,𝑥,𝑦   𝐷,𝑑,𝑓   𝑋,𝑑,𝑓,𝑛
Allowed substitution hints:   𝐷(𝑥,𝑦,𝑛)   𝐻(𝑛)   𝑋(𝑥,𝑦)

Proof of Theorem filnetlem4
Dummy variables 𝑘 𝑚 𝑡 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 filnet.h . . . . 5 𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)
2 filnet.d . . . . 5 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
31, 2filnetlem3 36363 . . . 4 (𝐻 = 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))
43simpri 485 . . 3 (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel))
54simprd 495 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ DirRel)
6 f2ndres 8038 . . . . 5 (2nd ↾ (𝐹 × 𝑋)):(𝐹 × 𝑋)⟶𝑋
74simpld 494 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ⊆ (𝐹 × 𝑋))
8 fssres2 6777 . . . . 5 (((2nd ↾ (𝐹 × 𝑋)):(𝐹 × 𝑋)⟶𝑋𝐻 ⊆ (𝐹 × 𝑋)) → (2nd𝐻):𝐻𝑋)
96, 7, 8sylancr 587 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (2nd𝐻):𝐻𝑋)
10 filtop 23879 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
11 xpexg 7769 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑋𝐹) → (𝐹 × 𝑋) ∈ V)
1210, 11mpdan 687 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (𝐹 × 𝑋) ∈ V)
1312, 7ssexd 5330 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ∈ V)
149, 13fexd 7247 . . 3 (𝐹 ∈ (Fil‘𝑋) → (2nd𝐻) ∈ V)
153simpli 483 . . . . . . 7 𝐻 = 𝐷
16 dirdm 18658 . . . . . . . 8 (𝐷 ∈ DirRel → dom 𝐷 = 𝐷)
175, 16syl 17 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → dom 𝐷 = 𝐷)
1815, 17eqtr4id 2794 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝐻 = dom 𝐷)
1918feq2d 6723 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ((2nd𝐻):𝐻𝑋 ↔ (2nd𝐻):dom 𝐷𝑋))
209, 19mpbid 232 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (2nd𝐻):dom 𝐷𝑋)
21 eqid 2735 . . . . . . . . . . . . . 14 dom 𝐷 = dom 𝐷
2221tailf 36358 . . . . . . . . . . . . 13 (𝐷 ∈ DirRel → (tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷)
235, 22syl 17 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → (tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷)
2418feq2d 6723 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → ((tail‘𝐷):𝐻⟶𝒫 dom 𝐷 ↔ (tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷))
2523, 24mpbird 257 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → (tail‘𝐷):𝐻⟶𝒫 dom 𝐷)
2625adantr 480 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (tail‘𝐷):𝐻⟶𝒫 dom 𝐷)
27 ffn 6737 . . . . . . . . . 10 ((tail‘𝐷):𝐻⟶𝒫 dom 𝐷 → (tail‘𝐷) Fn 𝐻)
28 imaeq2 6076 . . . . . . . . . . . 12 (𝑑 = ((tail‘𝐷)‘𝑓) → ((2nd𝐻) “ 𝑑) = ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)))
2928sseq1d 4027 . . . . . . . . . . 11 (𝑑 = ((tail‘𝐷)‘𝑓) → (((2nd𝐻) “ 𝑑) ⊆ 𝑡 ↔ ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡))
3029rexrn 7107 . . . . . . . . . 10 ((tail‘𝐷) Fn 𝐻 → (∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡 ↔ ∃𝑓𝐻 ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡))
3126, 27, 303syl 18 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡 ↔ ∃𝑓𝐻 ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡))
32 fo2nd 8034 . . . . . . . . . . . . . . 15 2nd :V–onto→V
33 fofn 6823 . . . . . . . . . . . . . . 15 (2nd :V–onto→V → 2nd Fn V)
3432, 33ax-mp 5 . . . . . . . . . . . . . 14 2nd Fn V
35 ssv 4020 . . . . . . . . . . . . . 14 𝐻 ⊆ V
36 fnssres 6692 . . . . . . . . . . . . . 14 ((2nd Fn V ∧ 𝐻 ⊆ V) → (2nd𝐻) Fn 𝐻)
3734, 35, 36mp2an 692 . . . . . . . . . . . . 13 (2nd𝐻) Fn 𝐻
38 fnfun 6669 . . . . . . . . . . . . 13 ((2nd𝐻) Fn 𝐻 → Fun (2nd𝐻))
3937, 38ax-mp 5 . . . . . . . . . . . 12 Fun (2nd𝐻)
4026ffvelcdmda 7104 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((tail‘𝐷)‘𝑓) ∈ 𝒫 dom 𝐷)
4140elpwid 4614 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((tail‘𝐷)‘𝑓) ⊆ dom 𝐷)
4218ad2antrr 726 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝐻 = dom 𝐷)
4341, 42sseqtrrd 4037 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((tail‘𝐷)‘𝑓) ⊆ 𝐻)
4437fndmi 6673 . . . . . . . . . . . . 13 dom (2nd𝐻) = 𝐻
4543, 44sseqtrrdi 4047 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((tail‘𝐷)‘𝑓) ⊆ dom (2nd𝐻))
46 funimass4 6973 . . . . . . . . . . . 12 ((Fun (2nd𝐻) ∧ ((tail‘𝐷)‘𝑓) ⊆ dom (2nd𝐻)) → (((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd𝐻)‘𝑑) ∈ 𝑡))
4739, 45, 46sylancr 587 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd𝐻)‘𝑑) ∈ 𝑡))
485ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝐷 ∈ DirRel)
49 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝑓𝐻)
5049, 42eleqtrd 2841 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝑓 ∈ dom 𝐷)
51 vex 3482 . . . . . . . . . . . . . . . . 17 𝑑 ∈ V
5251a1i 11 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝑑 ∈ V)
5321eltail 36357 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ 𝑓 ∈ dom 𝐷𝑑 ∈ V) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ 𝑓𝐷𝑑))
5448, 50, 52, 53syl3anc 1370 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ 𝑓𝐷𝑑))
5549biantrurd 532 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (𝑑𝐻 ↔ (𝑓𝐻𝑑𝐻)))
5655anbi1d 631 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) ↔ ((𝑓𝐻𝑑𝐻) ∧ (1st𝑑) ⊆ (1st𝑓))))
57 vex 3482 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
581, 2, 57, 51filnetlem1 36361 . . . . . . . . . . . . . . . 16 (𝑓𝐷𝑑 ↔ ((𝑓𝐻𝑑𝐻) ∧ (1st𝑑) ⊆ (1st𝑓)))
5956, 58bitr4di 289 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) ↔ 𝑓𝐷𝑑))
6054, 59bitr4d 282 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ (𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓))))
6160imbi1d 341 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((𝑑 ∈ ((tail‘𝐷)‘𝑓) → ((2nd𝐻)‘𝑑) ∈ 𝑡) ↔ ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → ((2nd𝐻)‘𝑑) ∈ 𝑡)))
62 fvres 6926 . . . . . . . . . . . . . . . . 17 (𝑑𝐻 → ((2nd𝐻)‘𝑑) = (2nd𝑑))
6362eleq1d 2824 . . . . . . . . . . . . . . . 16 (𝑑𝐻 → (((2nd𝐻)‘𝑑) ∈ 𝑡 ↔ (2nd𝑑) ∈ 𝑡))
6463adantr 480 . . . . . . . . . . . . . . 15 ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → (((2nd𝐻)‘𝑑) ∈ 𝑡 ↔ (2nd𝑑) ∈ 𝑡))
6564pm5.74i 271 . . . . . . . . . . . . . 14 (((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → ((2nd𝐻)‘𝑑) ∈ 𝑡) ↔ ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → (2nd𝑑) ∈ 𝑡))
66 impexp 450 . . . . . . . . . . . . . 14 (((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → (2nd𝑑) ∈ 𝑡) ↔ (𝑑𝐻 → ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
6765, 66bitri 275 . . . . . . . . . . . . 13 (((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → ((2nd𝐻)‘𝑑) ∈ 𝑡) ↔ (𝑑𝐻 → ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
6861, 67bitrdi 287 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((𝑑 ∈ ((tail‘𝐷)‘𝑓) → ((2nd𝐻)‘𝑑) ∈ 𝑡) ↔ (𝑑𝐻 → ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡))))
6968ralbidv2 3172 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd𝐻)‘𝑑) ∈ 𝑡 ↔ ∀𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
7047, 69bitrd 279 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
7170rexbidva 3175 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑓𝐻 ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∃𝑓𝐻𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
72 vex 3482 . . . . . . . . . . . . . . . . 17 𝑘 ∈ V
73 vex 3482 . . . . . . . . . . . . . . . . 17 𝑣 ∈ V
7472, 73op1std 8023 . . . . . . . . . . . . . . . 16 (𝑑 = ⟨𝑘, 𝑣⟩ → (1st𝑑) = 𝑘)
7574sseq1d 4027 . . . . . . . . . . . . . . 15 (𝑑 = ⟨𝑘, 𝑣⟩ → ((1st𝑑) ⊆ (1st𝑓) ↔ 𝑘 ⊆ (1st𝑓)))
7672, 73op2ndd 8024 . . . . . . . . . . . . . . . 16 (𝑑 = ⟨𝑘, 𝑣⟩ → (2nd𝑑) = 𝑣)
7776eleq1d 2824 . . . . . . . . . . . . . . 15 (𝑑 = ⟨𝑘, 𝑣⟩ → ((2nd𝑑) ∈ 𝑡𝑣𝑡))
7875, 77imbi12d 344 . . . . . . . . . . . . . 14 (𝑑 = ⟨𝑘, 𝑣⟩ → (((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ (𝑘 ⊆ (1st𝑓) → 𝑣𝑡)))
7978raliunxp 5853 . . . . . . . . . . . . 13 (∀𝑑 𝑘𝐹 ({𝑘} × 𝑘)((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∀𝑘𝐹𝑣𝑘 (𝑘 ⊆ (1st𝑓) → 𝑣𝑡))
80 sneq 4641 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → {𝑛} = {𝑘})
81 id 22 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘𝑛 = 𝑘)
8280, 81xpeq12d 5720 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → ({𝑛} × 𝑛) = ({𝑘} × 𝑘))
8382cbviunv 5045 . . . . . . . . . . . . . . 15 𝑛𝐹 ({𝑛} × 𝑛) = 𝑘𝐹 ({𝑘} × 𝑘)
841, 83eqtri 2763 . . . . . . . . . . . . . 14 𝐻 = 𝑘𝐹 ({𝑘} × 𝑘)
8584raleqi 3322 . . . . . . . . . . . . 13 (∀𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∀𝑑 𝑘𝐹 ({𝑘} × 𝑘)((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡))
86 dfss3 3984 . . . . . . . . . . . . . . . 16 (𝑘𝑡 ↔ ∀𝑣𝑘 𝑣𝑡)
8786imbi2i 336 . . . . . . . . . . . . . . 15 ((𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ (𝑘 ⊆ (1st𝑓) → ∀𝑣𝑘 𝑣𝑡))
88 r19.21v 3178 . . . . . . . . . . . . . . 15 (∀𝑣𝑘 (𝑘 ⊆ (1st𝑓) → 𝑣𝑡) ↔ (𝑘 ⊆ (1st𝑓) → ∀𝑣𝑘 𝑣𝑡))
8987, 88bitr4i 278 . . . . . . . . . . . . . 14 ((𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∀𝑣𝑘 (𝑘 ⊆ (1st𝑓) → 𝑣𝑡))
9089ralbii 3091 . . . . . . . . . . . . 13 (∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∀𝑘𝐹𝑣𝑘 (𝑘 ⊆ (1st𝑓) → 𝑣𝑡))
9179, 85, 903bitr4i 303 . . . . . . . . . . . 12 (∀𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡))
9291rexbii 3092 . . . . . . . . . . 11 (∃𝑓𝐻𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∃𝑓𝐻𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡))
931rexeqi 3323 . . . . . . . . . . 11 (∃𝑓𝐻𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∃𝑓 𝑛𝐹 ({𝑛} × 𝑛)∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡))
94 vex 3482 . . . . . . . . . . . . . . . 16 𝑛 ∈ V
95 vex 3482 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
9694, 95op1std 8023 . . . . . . . . . . . . . . 15 (𝑓 = ⟨𝑛, 𝑚⟩ → (1st𝑓) = 𝑛)
9796sseq2d 4028 . . . . . . . . . . . . . 14 (𝑓 = ⟨𝑛, 𝑚⟩ → (𝑘 ⊆ (1st𝑓) ↔ 𝑘𝑛))
9897imbi1d 341 . . . . . . . . . . . . 13 (𝑓 = ⟨𝑛, 𝑚⟩ → ((𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ (𝑘𝑛𝑘𝑡)))
9998ralbidv 3176 . . . . . . . . . . . 12 (𝑓 = ⟨𝑛, 𝑚⟩ → (∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∀𝑘𝐹 (𝑘𝑛𝑘𝑡)))
10099rexiunxp 5854 . . . . . . . . . . 11 (∃𝑓 𝑛𝐹 ({𝑛} × 𝑛)∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∃𝑛𝐹𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡))
10192, 93, 1003bitri 297 . . . . . . . . . 10 (∃𝑓𝐻𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∃𝑛𝐹𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡))
102 fileln0 23874 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → 𝑛 ≠ ∅)
103102adantlr 715 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → 𝑛 ≠ ∅)
104 r19.9rzv 4506 . . . . . . . . . . . . 13 (𝑛 ≠ ∅ → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ ∃𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡)))
105103, 104syl 17 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ ∃𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡)))
106 ssid 4018 . . . . . . . . . . . . . . 15 𝑛𝑛
107 sseq1 4021 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘𝑛𝑛𝑛))
108 sseq1 4021 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘𝑡𝑛𝑡))
109107, 108imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝑘𝑛𝑘𝑡) ↔ (𝑛𝑛𝑛𝑡)))
110109rspcv 3618 . . . . . . . . . . . . . . 15 (𝑛𝐹 → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) → (𝑛𝑛𝑛𝑡)))
111106, 110mpii 46 . . . . . . . . . . . . . 14 (𝑛𝐹 → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) → 𝑛𝑡))
112111adantl 481 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) → 𝑛𝑡))
113 sstr2 4002 . . . . . . . . . . . . . . 15 (𝑘𝑛 → (𝑛𝑡𝑘𝑡))
114113com12 32 . . . . . . . . . . . . . 14 (𝑛𝑡 → (𝑘𝑛𝑘𝑡))
115114ralrimivw 3148 . . . . . . . . . . . . 13 (𝑛𝑡 → ∀𝑘𝐹 (𝑘𝑛𝑘𝑡))
116112, 115impbid1 225 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ 𝑛𝑡))
117105, 116bitr3d 281 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → (∃𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ 𝑛𝑡))
118117rexbidva 3175 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑛𝐹𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ ∃𝑛𝐹 𝑛𝑡))
119101, 118bitrid 283 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑓𝐻𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∃𝑛𝐹 𝑛𝑡))
12031, 71, 1193bitrd 305 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡 ↔ ∃𝑛𝐹 𝑛𝑡))
121120pm5.32da 579 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → ((𝑡𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡) ↔ (𝑡𝑋 ∧ ∃𝑛𝐹 𝑛𝑡)))
122 filn0 23886 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
12394snnz 4781 . . . . . . . . . . . . . . . 16 {𝑛} ≠ ∅
124102, 123jctil 519 . . . . . . . . . . . . . . 15 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ({𝑛} ≠ ∅ ∧ 𝑛 ≠ ∅))
125 neanior 3033 . . . . . . . . . . . . . . 15 (({𝑛} ≠ ∅ ∧ 𝑛 ≠ ∅) ↔ ¬ ({𝑛} = ∅ ∨ 𝑛 = ∅))
126124, 125sylib 218 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ¬ ({𝑛} = ∅ ∨ 𝑛 = ∅))
127 ss0b 4407 . . . . . . . . . . . . . . 15 (({𝑛} × 𝑛) ⊆ ∅ ↔ ({𝑛} × 𝑛) = ∅)
128 xpeq0 6182 . . . . . . . . . . . . . . 15 (({𝑛} × 𝑛) = ∅ ↔ ({𝑛} = ∅ ∨ 𝑛 = ∅))
129127, 128bitri 275 . . . . . . . . . . . . . 14 (({𝑛} × 𝑛) ⊆ ∅ ↔ ({𝑛} = ∅ ∨ 𝑛 = ∅))
130126, 129sylnibr 329 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ¬ ({𝑛} × 𝑛) ⊆ ∅)
131130ralrimiva 3144 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → ∀𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅)
132 r19.2z 4501 . . . . . . . . . . . 12 ((𝐹 ≠ ∅ ∧ ∀𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅) → ∃𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅)
133122, 131, 132syl2anc 584 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → ∃𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅)
134 rexnal 3098 . . . . . . . . . . 11 (∃𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅ ↔ ¬ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
135133, 134sylib 218 . . . . . . . . . 10 (𝐹 ∈ (Fil‘𝑋) → ¬ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
1361sseq1i 4024 . . . . . . . . . . . 12 (𝐻 ⊆ ∅ ↔ 𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
137 ss0b 4407 . . . . . . . . . . . 12 (𝐻 ⊆ ∅ ↔ 𝐻 = ∅)
138 iunss 5050 . . . . . . . . . . . 12 ( 𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅ ↔ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
139136, 137, 1383bitr3i 301 . . . . . . . . . . 11 (𝐻 = ∅ ↔ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
140139necon3abii 2985 . . . . . . . . . 10 (𝐻 ≠ ∅ ↔ ¬ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
141135, 140sylibr 234 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ≠ ∅)
142 dmresi 6072 . . . . . . . . . . . 12 dom ( I ↾ 𝐻) = 𝐻
1431, 2filnetlem2 36362 . . . . . . . . . . . . . 14 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
144143simpli 483 . . . . . . . . . . . . 13 ( I ↾ 𝐻) ⊆ 𝐷
145 dmss 5916 . . . . . . . . . . . . 13 (( I ↾ 𝐻) ⊆ 𝐷 → dom ( I ↾ 𝐻) ⊆ dom 𝐷)
146144, 145ax-mp 5 . . . . . . . . . . . 12 dom ( I ↾ 𝐻) ⊆ dom 𝐷
147142, 146eqsstrri 4031 . . . . . . . . . . 11 𝐻 ⊆ dom 𝐷
148143simpri 485 . . . . . . . . . . . . 13 𝐷 ⊆ (𝐻 × 𝐻)
149 dmss 5916 . . . . . . . . . . . . 13 (𝐷 ⊆ (𝐻 × 𝐻) → dom 𝐷 ⊆ dom (𝐻 × 𝐻))
150148, 149ax-mp 5 . . . . . . . . . . . 12 dom 𝐷 ⊆ dom (𝐻 × 𝐻)
151 dmxpid 5944 . . . . . . . . . . . 12 dom (𝐻 × 𝐻) = 𝐻
152150, 151sseqtri 4032 . . . . . . . . . . 11 dom 𝐷𝐻
153147, 152eqssi 4012 . . . . . . . . . 10 𝐻 = dom 𝐷
154153tailfb 36360 . . . . . . . . 9 ((𝐷 ∈ DirRel ∧ 𝐻 ≠ ∅) → ran (tail‘𝐷) ∈ (fBas‘𝐻))
1555, 141, 154syl2anc 584 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → ran (tail‘𝐷) ∈ (fBas‘𝐻))
156 elfm 23971 . . . . . . . 8 ((𝑋𝐹 ∧ ran (tail‘𝐷) ∈ (fBas‘𝐻) ∧ (2nd𝐻):𝐻𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)) ↔ (𝑡𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡)))
15710, 155, 9, 156syl3anc 1370 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)) ↔ (𝑡𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡)))
158 filfbas 23872 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
159 elfg 23895 . . . . . . . 8 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑛𝐹 𝑛𝑡)))
160158, 159syl 17 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑛𝐹 𝑛𝑡)))
161121, 157, 1603bitr4d 311 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)) ↔ 𝑡 ∈ (𝑋filGen𝐹)))
162161eqrdv 2733 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)) = (𝑋filGen𝐹))
163 fgfil 23899 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
164162, 163eqtr2d 2776 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)))
16520, 164jca 511 . . 3 (𝐹 ∈ (Fil‘𝑋) → ((2nd𝐻):dom 𝐷𝑋𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷))))
166 feq1 6717 . . . . 5 (𝑓 = (2nd𝐻) → (𝑓:dom 𝐷𝑋 ↔ (2nd𝐻):dom 𝐷𝑋))
167 oveq2 7439 . . . . . . 7 (𝑓 = (2nd𝐻) → (𝑋 FilMap 𝑓) = (𝑋 FilMap (2nd𝐻)))
168167fveq1d 6909 . . . . . 6 (𝑓 = (2nd𝐻) → ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)) = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)))
169168eqeq2d 2746 . . . . 5 (𝑓 = (2nd𝐻) → (𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)) ↔ 𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷))))
170166, 169anbi12d 632 . . . 4 (𝑓 = (2nd𝐻) → ((𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))) ↔ ((2nd𝐻):dom 𝐷𝑋𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)))))
171170spcegv 3597 . . 3 ((2nd𝐻) ∈ V → (((2nd𝐻):dom 𝐷𝑋𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷))) → ∃𝑓(𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))))
17214, 165, 171sylc 65 . 2 (𝐹 ∈ (Fil‘𝑋) → ∃𝑓(𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))))
173 dmeq 5917 . . . . . 6 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
174173feq2d 6723 . . . . 5 (𝑑 = 𝐷 → (𝑓:dom 𝑑𝑋𝑓:dom 𝐷𝑋))
175 fveq2 6907 . . . . . . . 8 (𝑑 = 𝐷 → (tail‘𝑑) = (tail‘𝐷))
176175rneqd 5952 . . . . . . 7 (𝑑 = 𝐷 → ran (tail‘𝑑) = ran (tail‘𝐷))
177176fveq2d 6911 . . . . . 6 (𝑑 = 𝐷 → ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)) = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))
178177eqeq2d 2746 . . . . 5 (𝑑 = 𝐷 → (𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)) ↔ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))))
179174, 178anbi12d 632 . . . 4 (𝑑 = 𝐷 → ((𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))) ↔ (𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))))
180179exbidv 1919 . . 3 (𝑑 = 𝐷 → (∃𝑓(𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))) ↔ ∃𝑓(𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))))
181180rspcev 3622 . 2 ((𝐷 ∈ DirRel ∧ ∃𝑓(𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))
1825, 172, 181syl2anc 584 1 (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631  cop 4637   cuni 4912   ciun 4996   class class class wbr 5148  {copab 5210   I cid 5582   × cxp 5687  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Fun wfun 6557   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  DirRelcdir 18652  tailctail 18653  fBascfbas 21370  filGencfg 21371  Filcfil 23869   FilMap cfm 23957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-dir 18654  df-tail 18655  df-fbas 21379  df-fg 21380  df-fil 23870  df-fm 23962
This theorem is referenced by:  filnet  36365
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