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Theorem filnetlem4 34853
Description: Lemma for filnet 34854. (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 34852 . . . 4 (𝐻 = 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))
43simpri 486 . . 3 (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel))
54simprd 496 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ DirRel)
6 f2ndres 7946 . . . . 5 (2nd ↾ (𝐹 × 𝑋)):(𝐹 × 𝑋)⟶𝑋
74simpld 495 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ⊆ (𝐹 × 𝑋))
8 fssres2 6710 . . . . 5 (((2nd ↾ (𝐹 × 𝑋)):(𝐹 × 𝑋)⟶𝑋𝐻 ⊆ (𝐹 × 𝑋)) → (2nd𝐻):𝐻𝑋)
96, 7, 8sylancr 587 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (2nd𝐻):𝐻𝑋)
10 filtop 23206 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
11 xpexg 7684 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑋𝐹) → (𝐹 × 𝑋) ∈ V)
1210, 11mpdan 685 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (𝐹 × 𝑋) ∈ V)
1312, 7ssexd 5281 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ∈ V)
149, 13fexd 7177 . . 3 (𝐹 ∈ (Fil‘𝑋) → (2nd𝐻) ∈ V)
153simpli 484 . . . . . . 7 𝐻 = 𝐷
16 dirdm 18489 . . . . . . . 8 (𝐷 ∈ DirRel → dom 𝐷 = 𝐷)
175, 16syl 17 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → dom 𝐷 = 𝐷)
1815, 17eqtr4id 2795 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝐻 = dom 𝐷)
1918feq2d 6654 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ((2nd𝐻):𝐻𝑋 ↔ (2nd𝐻):dom 𝐷𝑋))
209, 19mpbid 231 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (2nd𝐻):dom 𝐷𝑋)
21 eqid 2736 . . . . . . . . . . . . . 14 dom 𝐷 = dom 𝐷
2221tailf 34847 . . . . . . . . . . . . 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 6654 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → ((tail‘𝐷):𝐻⟶𝒫 dom 𝐷 ↔ (tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷))
2523, 24mpbird 256 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → (tail‘𝐷):𝐻⟶𝒫 dom 𝐷)
2625adantr 481 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (tail‘𝐷):𝐻⟶𝒫 dom 𝐷)
27 ffn 6668 . . . . . . . . . 10 ((tail‘𝐷):𝐻⟶𝒫 dom 𝐷 → (tail‘𝐷) Fn 𝐻)
28 imaeq2 6009 . . . . . . . . . . . 12 (𝑑 = ((tail‘𝐷)‘𝑓) → ((2nd𝐻) “ 𝑑) = ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)))
2928sseq1d 3975 . . . . . . . . . . 11 (𝑑 = ((tail‘𝐷)‘𝑓) → (((2nd𝐻) “ 𝑑) ⊆ 𝑡 ↔ ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡))
3029rexrn 7037 . . . . . . . . . 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 7942 . . . . . . . . . . . . . . 15 2nd :V–onto→V
33 fofn 6758 . . . . . . . . . . . . . . 15 (2nd :V–onto→V → 2nd Fn V)
3432, 33ax-mp 5 . . . . . . . . . . . . . 14 2nd Fn V
35 ssv 3968 . . . . . . . . . . . . . 14 𝐻 ⊆ V
36 fnssres 6624 . . . . . . . . . . . . . 14 ((2nd Fn V ∧ 𝐻 ⊆ V) → (2nd𝐻) Fn 𝐻)
3734, 35, 36mp2an 690 . . . . . . . . . . . . 13 (2nd𝐻) Fn 𝐻
38 fnfun 6602 . . . . . . . . . . . . 13 ((2nd𝐻) Fn 𝐻 → Fun (2nd𝐻))
3937, 38ax-mp 5 . . . . . . . . . . . 12 Fun (2nd𝐻)
4026ffvelcdmda 7035 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((tail‘𝐷)‘𝑓) ∈ 𝒫 dom 𝐷)
4140elpwid 4569 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((tail‘𝐷)‘𝑓) ⊆ dom 𝐷)
4218ad2antrr 724 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝐻 = dom 𝐷)
4341, 42sseqtrrd 3985 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((tail‘𝐷)‘𝑓) ⊆ 𝐻)
4437fndmi 6606 . . . . . . . . . . . . 13 dom (2nd𝐻) = 𝐻
4543, 44sseqtrrdi 3995 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((tail‘𝐷)‘𝑓) ⊆ dom (2nd𝐻))
46 funimass4 6907 . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝐷 ∈ DirRel)
49 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝑓𝐻)
5049, 42eleqtrd 2840 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝑓 ∈ dom 𝐷)
51 vex 3449 . . . . . . . . . . . . . . . . 17 𝑑 ∈ V
5251a1i 11 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝑑 ∈ V)
5321eltail 34846 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ 𝑓 ∈ dom 𝐷𝑑 ∈ V) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ 𝑓𝐷𝑑))
5448, 50, 52, 53syl3anc 1371 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ 𝑓𝐷𝑑))
5549biantrurd 533 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (𝑑𝐻 ↔ (𝑓𝐻𝑑𝐻)))
5655anbi1d 630 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) ↔ ((𝑓𝐻𝑑𝐻) ∧ (1st𝑑) ⊆ (1st𝑓))))
57 vex 3449 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
581, 2, 57, 51filnetlem1 34850 . . . . . . . . . . . . . . . 16 (𝑓𝐷𝑑 ↔ ((𝑓𝐻𝑑𝐻) ∧ (1st𝑑) ⊆ (1st𝑓)))
5956, 58bitr4di 288 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) ↔ 𝑓𝐷𝑑))
6054, 59bitr4d 281 . . . . . . . . . . . . . 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 6861 . . . . . . . . . . . . . . . . 17 (𝑑𝐻 → ((2nd𝐻)‘𝑑) = (2nd𝑑))
6362eleq1d 2822 . . . . . . . . . . . . . . . 16 (𝑑𝐻 → (((2nd𝐻)‘𝑑) ∈ 𝑡 ↔ (2nd𝑑) ∈ 𝑡))
6463adantr 481 . . . . . . . . . . . . . . 15 ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → (((2nd𝐻)‘𝑑) ∈ 𝑡 ↔ (2nd𝑑) ∈ 𝑡))
6564pm5.74i 270 . . . . . . . . . . . . . 14 (((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → ((2nd𝐻)‘𝑑) ∈ 𝑡) ↔ ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → (2nd𝑑) ∈ 𝑡))
66 impexp 451 . . . . . . . . . . . . . 14 (((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → (2nd𝑑) ∈ 𝑡) ↔ (𝑑𝐻 → ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
6765, 66bitri 274 . . . . . . . . . . . . 13 (((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → ((2nd𝐻)‘𝑑) ∈ 𝑡) ↔ (𝑑𝐻 → ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
6861, 67bitrdi 286 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((𝑑 ∈ ((tail‘𝐷)‘𝑓) → ((2nd𝐻)‘𝑑) ∈ 𝑡) ↔ (𝑑𝐻 → ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡))))
6968ralbidv2 3170 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd𝐻)‘𝑑) ∈ 𝑡 ↔ ∀𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
7047, 69bitrd 278 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
7170rexbidva 3173 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑓𝐻 ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∃𝑓𝐻𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
72 vex 3449 . . . . . . . . . . . . . . . . 17 𝑘 ∈ V
73 vex 3449 . . . . . . . . . . . . . . . . 17 𝑣 ∈ V
7472, 73op1std 7931 . . . . . . . . . . . . . . . 16 (𝑑 = ⟨𝑘, 𝑣⟩ → (1st𝑑) = 𝑘)
7574sseq1d 3975 . . . . . . . . . . . . . . 15 (𝑑 = ⟨𝑘, 𝑣⟩ → ((1st𝑑) ⊆ (1st𝑓) ↔ 𝑘 ⊆ (1st𝑓)))
7672, 73op2ndd 7932 . . . . . . . . . . . . . . . 16 (𝑑 = ⟨𝑘, 𝑣⟩ → (2nd𝑑) = 𝑣)
7776eleq1d 2822 . . . . . . . . . . . . . . 15 (𝑑 = ⟨𝑘, 𝑣⟩ → ((2nd𝑑) ∈ 𝑡𝑣𝑡))
7875, 77imbi12d 344 . . . . . . . . . . . . . 14 (𝑑 = ⟨𝑘, 𝑣⟩ → (((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ (𝑘 ⊆ (1st𝑓) → 𝑣𝑡)))
7978raliunxp 5795 . . . . . . . . . . . . 13 (∀𝑑 𝑘𝐹 ({𝑘} × 𝑘)((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∀𝑘𝐹𝑣𝑘 (𝑘 ⊆ (1st𝑓) → 𝑣𝑡))
80 sneq 4596 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → {𝑛} = {𝑘})
81 id 22 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘𝑛 = 𝑘)
8280, 81xpeq12d 5664 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → ({𝑛} × 𝑛) = ({𝑘} × 𝑘))
8382cbviunv 5000 . . . . . . . . . . . . . . 15 𝑛𝐹 ({𝑛} × 𝑛) = 𝑘𝐹 ({𝑘} × 𝑘)
841, 83eqtri 2764 . . . . . . . . . . . . . 14 𝐻 = 𝑘𝐹 ({𝑘} × 𝑘)
8584raleqi 3311 . . . . . . . . . . . . 13 (∀𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∀𝑑 𝑘𝐹 ({𝑘} × 𝑘)((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡))
86 dfss3 3932 . . . . . . . . . . . . . . . 16 (𝑘𝑡 ↔ ∀𝑣𝑘 𝑣𝑡)
8786imbi2i 335 . . . . . . . . . . . . . . 15 ((𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ (𝑘 ⊆ (1st𝑓) → ∀𝑣𝑘 𝑣𝑡))
88 r19.21v 3176 . . . . . . . . . . . . . . 15 (∀𝑣𝑘 (𝑘 ⊆ (1st𝑓) → 𝑣𝑡) ↔ (𝑘 ⊆ (1st𝑓) → ∀𝑣𝑘 𝑣𝑡))
8987, 88bitr4i 277 . . . . . . . . . . . . . 14 ((𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∀𝑣𝑘 (𝑘 ⊆ (1st𝑓) → 𝑣𝑡))
9089ralbii 3096 . . . . . . . . . . . . 13 (∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∀𝑘𝐹𝑣𝑘 (𝑘 ⊆ (1st𝑓) → 𝑣𝑡))
9179, 85, 903bitr4i 302 . . . . . . . . . . . 12 (∀𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡))
9291rexbii 3097 . . . . . . . . . . 11 (∃𝑓𝐻𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∃𝑓𝐻𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡))
931rexeqi 3312 . . . . . . . . . . 11 (∃𝑓𝐻𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∃𝑓 𝑛𝐹 ({𝑛} × 𝑛)∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡))
94 vex 3449 . . . . . . . . . . . . . . . 16 𝑛 ∈ V
95 vex 3449 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
9694, 95op1std 7931 . . . . . . . . . . . . . . 15 (𝑓 = ⟨𝑛, 𝑚⟩ → (1st𝑓) = 𝑛)
9796sseq2d 3976 . . . . . . . . . . . . . 14 (𝑓 = ⟨𝑛, 𝑚⟩ → (𝑘 ⊆ (1st𝑓) ↔ 𝑘𝑛))
9897imbi1d 341 . . . . . . . . . . . . 13 (𝑓 = ⟨𝑛, 𝑚⟩ → ((𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ (𝑘𝑛𝑘𝑡)))
9998ralbidv 3174 . . . . . . . . . . . 12 (𝑓 = ⟨𝑛, 𝑚⟩ → (∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∀𝑘𝐹 (𝑘𝑛𝑘𝑡)))
10099rexiunxp 5796 . . . . . . . . . . 11 (∃𝑓 𝑛𝐹 ({𝑛} × 𝑛)∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∃𝑛𝐹𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡))
10192, 93, 1003bitri 296 . . . . . . . . . 10 (∃𝑓𝐻𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∃𝑛𝐹𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡))
102 fileln0 23201 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → 𝑛 ≠ ∅)
103102adantlr 713 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → 𝑛 ≠ ∅)
104 r19.9rzv 4457 . . . . . . . . . . . . 13 (𝑛 ≠ ∅ → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ ∃𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡)))
105103, 104syl 17 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ ∃𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡)))
106 ssid 3966 . . . . . . . . . . . . . . 15 𝑛𝑛
107 sseq1 3969 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘𝑛𝑛𝑛))
108 sseq1 3969 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘𝑡𝑛𝑡))
109107, 108imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝑘𝑛𝑘𝑡) ↔ (𝑛𝑛𝑛𝑡)))
110109rspcv 3577 . . . . . . . . . . . . . . 15 (𝑛𝐹 → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) → (𝑛𝑛𝑛𝑡)))
111106, 110mpii 46 . . . . . . . . . . . . . 14 (𝑛𝐹 → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) → 𝑛𝑡))
112111adantl 482 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) → 𝑛𝑡))
113 sstr2 3951 . . . . . . . . . . . . . . 15 (𝑘𝑛 → (𝑛𝑡𝑘𝑡))
114113com12 32 . . . . . . . . . . . . . 14 (𝑛𝑡 → (𝑘𝑛𝑘𝑡))
115114ralrimivw 3147 . . . . . . . . . . . . 13 (𝑛𝑡 → ∀𝑘𝐹 (𝑘𝑛𝑘𝑡))
116112, 115impbid1 224 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ 𝑛𝑡))
117105, 116bitr3d 280 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → (∃𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ 𝑛𝑡))
118117rexbidva 3173 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑛𝐹𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ ∃𝑛𝐹 𝑛𝑡))
119101, 118bitrid 282 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑓𝐻𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∃𝑛𝐹 𝑛𝑡))
12031, 71, 1193bitrd 304 . . . . . . . 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 23213 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
12394snnz 4737 . . . . . . . . . . . . . . . 16 {𝑛} ≠ ∅
124102, 123jctil 520 . . . . . . . . . . . . . . 15 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ({𝑛} ≠ ∅ ∧ 𝑛 ≠ ∅))
125 neanior 3037 . . . . . . . . . . . . . . 15 (({𝑛} ≠ ∅ ∧ 𝑛 ≠ ∅) ↔ ¬ ({𝑛} = ∅ ∨ 𝑛 = ∅))
126124, 125sylib 217 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ¬ ({𝑛} = ∅ ∨ 𝑛 = ∅))
127 ss0b 4357 . . . . . . . . . . . . . . 15 (({𝑛} × 𝑛) ⊆ ∅ ↔ ({𝑛} × 𝑛) = ∅)
128 xpeq0 6112 . . . . . . . . . . . . . . 15 (({𝑛} × 𝑛) = ∅ ↔ ({𝑛} = ∅ ∨ 𝑛 = ∅))
129127, 128bitri 274 . . . . . . . . . . . . . 14 (({𝑛} × 𝑛) ⊆ ∅ ↔ ({𝑛} = ∅ ∨ 𝑛 = ∅))
130126, 129sylnibr 328 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ¬ ({𝑛} × 𝑛) ⊆ ∅)
131130ralrimiva 3143 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → ∀𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅)
132 r19.2z 4452 . . . . . . . . . . . 12 ((𝐹 ≠ ∅ ∧ ∀𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅) → ∃𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅)
133122, 131, 132syl2anc 584 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → ∃𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅)
134 rexnal 3103 . . . . . . . . . . 11 (∃𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅ ↔ ¬ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
135133, 134sylib 217 . . . . . . . . . 10 (𝐹 ∈ (Fil‘𝑋) → ¬ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
1361sseq1i 3972 . . . . . . . . . . . 12 (𝐻 ⊆ ∅ ↔ 𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
137 ss0b 4357 . . . . . . . . . . . 12 (𝐻 ⊆ ∅ ↔ 𝐻 = ∅)
138 iunss 5005 . . . . . . . . . . . 12 ( 𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅ ↔ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
139136, 137, 1383bitr3i 300 . . . . . . . . . . 11 (𝐻 = ∅ ↔ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
140139necon3abii 2990 . . . . . . . . . 10 (𝐻 ≠ ∅ ↔ ¬ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
141135, 140sylibr 233 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ≠ ∅)
142 dmresi 6005 . . . . . . . . . . . 12 dom ( I ↾ 𝐻) = 𝐻
1431, 2filnetlem2 34851 . . . . . . . . . . . . . 14 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
144143simpli 484 . . . . . . . . . . . . 13 ( I ↾ 𝐻) ⊆ 𝐷
145 dmss 5858 . . . . . . . . . . . . 13 (( I ↾ 𝐻) ⊆ 𝐷 → dom ( I ↾ 𝐻) ⊆ dom 𝐷)
146144, 145ax-mp 5 . . . . . . . . . . . 12 dom ( I ↾ 𝐻) ⊆ dom 𝐷
147142, 146eqsstrri 3979 . . . . . . . . . . 11 𝐻 ⊆ dom 𝐷
148143simpri 486 . . . . . . . . . . . . 13 𝐷 ⊆ (𝐻 × 𝐻)
149 dmss 5858 . . . . . . . . . . . . 13 (𝐷 ⊆ (𝐻 × 𝐻) → dom 𝐷 ⊆ dom (𝐻 × 𝐻))
150148, 149ax-mp 5 . . . . . . . . . . . 12 dom 𝐷 ⊆ dom (𝐻 × 𝐻)
151 dmxpid 5885 . . . . . . . . . . . 12 dom (𝐻 × 𝐻) = 𝐻
152150, 151sseqtri 3980 . . . . . . . . . . 11 dom 𝐷𝐻
153147, 152eqssi 3960 . . . . . . . . . 10 𝐻 = dom 𝐷
154153tailfb 34849 . . . . . . . . 9 ((𝐷 ∈ DirRel ∧ 𝐻 ≠ ∅) → ran (tail‘𝐷) ∈ (fBas‘𝐻))
1555, 141, 154syl2anc 584 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → ran (tail‘𝐷) ∈ (fBas‘𝐻))
156 elfm 23298 . . . . . . . 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 1371 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)) ↔ (𝑡𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡)))
158 filfbas 23199 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
159 elfg 23222 . . . . . . . 8 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑛𝐹 𝑛𝑡)))
160158, 159syl 17 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑛𝐹 𝑛𝑡)))
161121, 157, 1603bitr4d 310 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)) ↔ 𝑡 ∈ (𝑋filGen𝐹)))
162161eqrdv 2734 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)) = (𝑋filGen𝐹))
163 fgfil 23226 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
164162, 163eqtr2d 2777 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)))
16520, 164jca 512 . . 3 (𝐹 ∈ (Fil‘𝑋) → ((2nd𝐻):dom 𝐷𝑋𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷))))
166 feq1 6649 . . . . 5 (𝑓 = (2nd𝐻) → (𝑓:dom 𝐷𝑋 ↔ (2nd𝐻):dom 𝐷𝑋))
167 oveq2 7365 . . . . . . 7 (𝑓 = (2nd𝐻) → (𝑋 FilMap 𝑓) = (𝑋 FilMap (2nd𝐻)))
168167fveq1d 6844 . . . . . 6 (𝑓 = (2nd𝐻) → ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)) = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)))
169168eqeq2d 2747 . . . . 5 (𝑓 = (2nd𝐻) → (𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)) ↔ 𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷))))
170166, 169anbi12d 631 . . . 4 (𝑓 = (2nd𝐻) → ((𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))) ↔ ((2nd𝐻):dom 𝐷𝑋𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)))))
171170spcegv 3556 . . 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 5859 . . . . . 6 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
174173feq2d 6654 . . . . 5 (𝑑 = 𝐷 → (𝑓:dom 𝑑𝑋𝑓:dom 𝐷𝑋))
175 fveq2 6842 . . . . . . . 8 (𝑑 = 𝐷 → (tail‘𝑑) = (tail‘𝐷))
176175rneqd 5893 . . . . . . 7 (𝑑 = 𝐷 → ran (tail‘𝑑) = ran (tail‘𝐷))
177176fveq2d 6846 . . . . . 6 (𝑑 = 𝐷 → ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)) = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))
178177eqeq2d 2747 . . . . 5 (𝑑 = 𝐷 → (𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)) ↔ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))))
179174, 178anbi12d 631 . . . 4 (𝑑 = 𝐷 → ((𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))) ↔ (𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))))
180179exbidv 1924 . . 3 (𝑑 = 𝐷 → (∃𝑓(𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))) ↔ ∃𝑓(𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))))
181180rspcev 3581 . 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 205  wa 396  wo 845   = wceq 1541  wex 1781  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586  cop 4592   cuni 4865   ciun 4954   class class class wbr 5105  {copab 5167   I cid 5530   × cxp 5631  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  DirRelcdir 18483  tailctail 18484  fBascfbas 20784  filGencfg 20785  Filcfil 23196   FilMap cfm 23284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-dir 18485  df-tail 18486  df-fbas 20793  df-fg 20794  df-fil 23197  df-fm 23289
This theorem is referenced by:  filnet  34854
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