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Theorem filnetlem4 32045
Description: Lemma for filnet 32046. (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 32044 . . . 4 (𝐻 = 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))
43simpri 478 . . 3 (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel))
54simprd 479 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ DirRel)
6 f2ndres 7143 . . . . 5 (2nd ↾ (𝐹 × 𝑋)):(𝐹 × 𝑋)⟶𝑋
74simpld 475 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ⊆ (𝐹 × 𝑋))
8 fssres2 6034 . . . . 5 (((2nd ↾ (𝐹 × 𝑋)):(𝐹 × 𝑋)⟶𝑋𝐻 ⊆ (𝐹 × 𝑋)) → (2nd𝐻):𝐻𝑋)
96, 7, 8sylancr 694 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (2nd𝐻):𝐻𝑋)
10 filtop 21578 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
11 xpexg 6920 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑋𝐹) → (𝐹 × 𝑋) ∈ V)
1210, 11mpdan 701 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (𝐹 × 𝑋) ∈ V)
1312, 7ssexd 4770 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ∈ V)
14 fex 6450 . . . 4 (((2nd𝐻):𝐻𝑋𝐻 ∈ V) → (2nd𝐻) ∈ V)
159, 13, 14syl2anc 692 . . 3 (𝐹 ∈ (Fil‘𝑋) → (2nd𝐻) ∈ V)
16 dirdm 17162 . . . . . . . 8 (𝐷 ∈ DirRel → dom 𝐷 = 𝐷)
175, 16syl 17 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → dom 𝐷 = 𝐷)
183simpli 474 . . . . . . 7 𝐻 = 𝐷
1917, 18syl6reqr 2674 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝐻 = dom 𝐷)
2019feq2d 5993 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ((2nd𝐻):𝐻𝑋 ↔ (2nd𝐻):dom 𝐷𝑋))
219, 20mpbid 222 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (2nd𝐻):dom 𝐷𝑋)
22 eqid 2621 . . . . . . . . . . . . . 14 dom 𝐷 = dom 𝐷
2322tailf 32039 . . . . . . . . . . . . 13 (𝐷 ∈ DirRel → (tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷)
245, 23syl 17 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → (tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷)
2519feq2d 5993 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → ((tail‘𝐷):𝐻⟶𝒫 dom 𝐷 ↔ (tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷))
2624, 25mpbird 247 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → (tail‘𝐷):𝐻⟶𝒫 dom 𝐷)
2726adantr 481 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (tail‘𝐷):𝐻⟶𝒫 dom 𝐷)
28 ffn 6007 . . . . . . . . . 10 ((tail‘𝐷):𝐻⟶𝒫 dom 𝐷 → (tail‘𝐷) Fn 𝐻)
29 imaeq2 5426 . . . . . . . . . . . 12 (𝑑 = ((tail‘𝐷)‘𝑓) → ((2nd𝐻) “ 𝑑) = ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)))
3029sseq1d 3616 . . . . . . . . . . 11 (𝑑 = ((tail‘𝐷)‘𝑓) → (((2nd𝐻) “ 𝑑) ⊆ 𝑡 ↔ ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡))
3130rexrn 6322 . . . . . . . . . 10 ((tail‘𝐷) Fn 𝐻 → (∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡 ↔ ∃𝑓𝐻 ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡))
3227, 28, 313syl 18 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡 ↔ ∃𝑓𝐻 ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡))
33 fo2nd 7141 . . . . . . . . . . . . . . 15 2nd :V–onto→V
34 fofn 6079 . . . . . . . . . . . . . . 15 (2nd :V–onto→V → 2nd Fn V)
3533, 34ax-mp 5 . . . . . . . . . . . . . 14 2nd Fn V
36 ssv 3609 . . . . . . . . . . . . . 14 𝐻 ⊆ V
37 fnssres 5967 . . . . . . . . . . . . . 14 ((2nd Fn V ∧ 𝐻 ⊆ V) → (2nd𝐻) Fn 𝐻)
3835, 36, 37mp2an 707 . . . . . . . . . . . . 13 (2nd𝐻) Fn 𝐻
39 fnfun 5951 . . . . . . . . . . . . 13 ((2nd𝐻) Fn 𝐻 → Fun (2nd𝐻))
4038, 39ax-mp 5 . . . . . . . . . . . 12 Fun (2nd𝐻)
4127ffvelrnda 6320 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((tail‘𝐷)‘𝑓) ∈ 𝒫 dom 𝐷)
4241elpwid 4146 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((tail‘𝐷)‘𝑓) ⊆ dom 𝐷)
4319ad2antrr 761 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝐻 = dom 𝐷)
4442, 43sseqtr4d 3626 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((tail‘𝐷)‘𝑓) ⊆ 𝐻)
45 fndm 5953 . . . . . . . . . . . . . 14 ((2nd𝐻) Fn 𝐻 → dom (2nd𝐻) = 𝐻)
4638, 45ax-mp 5 . . . . . . . . . . . . 13 dom (2nd𝐻) = 𝐻
4744, 46syl6sseqr 3636 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((tail‘𝐷)‘𝑓) ⊆ dom (2nd𝐻))
48 funimass4 6209 . . . . . . . . . . . 12 ((Fun (2nd𝐻) ∧ ((tail‘𝐷)‘𝑓) ⊆ dom (2nd𝐻)) → (((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd𝐻)‘𝑑) ∈ 𝑡))
4940, 47, 48sylancr 694 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd𝐻)‘𝑑) ∈ 𝑡))
505ad2antrr 761 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝐷 ∈ DirRel)
51 simpr 477 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝑓𝐻)
5251, 43eleqtrd 2700 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝑓 ∈ dom 𝐷)
53 vex 3192 . . . . . . . . . . . . . . . . 17 𝑑 ∈ V
5453a1i 11 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → 𝑑 ∈ V)
5522eltail 32038 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ 𝑓 ∈ dom 𝐷𝑑 ∈ V) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ 𝑓𝐷𝑑))
5650, 52, 54, 55syl3anc 1323 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ 𝑓𝐷𝑑))
5751biantrurd 529 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (𝑑𝐻 ↔ (𝑓𝐻𝑑𝐻)))
5857anbi1d 740 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) ↔ ((𝑓𝐻𝑑𝐻) ∧ (1st𝑑) ⊆ (1st𝑓))))
59 vex 3192 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
601, 2, 59, 53filnetlem1 32042 . . . . . . . . . . . . . . . 16 (𝑓𝐷𝑑 ↔ ((𝑓𝐻𝑑𝐻) ∧ (1st𝑑) ⊆ (1st𝑓)))
6158, 60syl6bbr 278 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) ↔ 𝑓𝐷𝑑))
6256, 61bitr4d 271 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ (𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓))))
6362imbi1d 331 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((𝑑 ∈ ((tail‘𝐷)‘𝑓) → ((2nd𝐻)‘𝑑) ∈ 𝑡) ↔ ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → ((2nd𝐻)‘𝑑) ∈ 𝑡)))
64 fvres 6169 . . . . . . . . . . . . . . . . 17 (𝑑𝐻 → ((2nd𝐻)‘𝑑) = (2nd𝑑))
6564eleq1d 2683 . . . . . . . . . . . . . . . 16 (𝑑𝐻 → (((2nd𝐻)‘𝑑) ∈ 𝑡 ↔ (2nd𝑑) ∈ 𝑡))
6665adantr 481 . . . . . . . . . . . . . . 15 ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → (((2nd𝐻)‘𝑑) ∈ 𝑡 ↔ (2nd𝑑) ∈ 𝑡))
6766pm5.74i 260 . . . . . . . . . . . . . 14 (((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → ((2nd𝐻)‘𝑑) ∈ 𝑡) ↔ ((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → (2nd𝑑) ∈ 𝑡))
68 impexp 462 . . . . . . . . . . . . . 14 (((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → (2nd𝑑) ∈ 𝑡) ↔ (𝑑𝐻 → ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
6967, 68bitri 264 . . . . . . . . . . . . 13 (((𝑑𝐻 ∧ (1st𝑑) ⊆ (1st𝑓)) → ((2nd𝐻)‘𝑑) ∈ 𝑡) ↔ (𝑑𝐻 → ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
7063, 69syl6bb 276 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → ((𝑑 ∈ ((tail‘𝐷)‘𝑓) → ((2nd𝐻)‘𝑑) ∈ 𝑡) ↔ (𝑑𝐻 → ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡))))
7170ralbidv2 2979 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd𝐻)‘𝑑) ∈ 𝑡 ↔ ∀𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
7249, 71bitrd 268 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑓𝐻) → (((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
7372rexbidva 3043 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑓𝐻 ((2nd𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∃𝑓𝐻𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡)))
74 vex 3192 . . . . . . . . . . . . . . . . 17 𝑘 ∈ V
75 vex 3192 . . . . . . . . . . . . . . . . 17 𝑣 ∈ V
7674, 75op1std 7130 . . . . . . . . . . . . . . . 16 (𝑑 = ⟨𝑘, 𝑣⟩ → (1st𝑑) = 𝑘)
7776sseq1d 3616 . . . . . . . . . . . . . . 15 (𝑑 = ⟨𝑘, 𝑣⟩ → ((1st𝑑) ⊆ (1st𝑓) ↔ 𝑘 ⊆ (1st𝑓)))
7874, 75op2ndd 7131 . . . . . . . . . . . . . . . 16 (𝑑 = ⟨𝑘, 𝑣⟩ → (2nd𝑑) = 𝑣)
7978eleq1d 2683 . . . . . . . . . . . . . . 15 (𝑑 = ⟨𝑘, 𝑣⟩ → ((2nd𝑑) ∈ 𝑡𝑣𝑡))
8077, 79imbi12d 334 . . . . . . . . . . . . . 14 (𝑑 = ⟨𝑘, 𝑣⟩ → (((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ (𝑘 ⊆ (1st𝑓) → 𝑣𝑡)))
8180raliunxp 5226 . . . . . . . . . . . . 13 (∀𝑑 𝑘𝐹 ({𝑘} × 𝑘)((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∀𝑘𝐹𝑣𝑘 (𝑘 ⊆ (1st𝑓) → 𝑣𝑡))
82 sneq 4163 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → {𝑛} = {𝑘})
83 id 22 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘𝑛 = 𝑘)
8482, 83xpeq12d 5105 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → ({𝑛} × 𝑛) = ({𝑘} × 𝑘))
8584cbviunv 4530 . . . . . . . . . . . . . . 15 𝑛𝐹 ({𝑛} × 𝑛) = 𝑘𝐹 ({𝑘} × 𝑘)
861, 85eqtri 2643 . . . . . . . . . . . . . 14 𝐻 = 𝑘𝐹 ({𝑘} × 𝑘)
8786raleqi 3134 . . . . . . . . . . . . 13 (∀𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∀𝑑 𝑘𝐹 ({𝑘} × 𝑘)((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡))
88 dfss3 3577 . . . . . . . . . . . . . . . 16 (𝑘𝑡 ↔ ∀𝑣𝑘 𝑣𝑡)
8988imbi2i 326 . . . . . . . . . . . . . . 15 ((𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ (𝑘 ⊆ (1st𝑓) → ∀𝑣𝑘 𝑣𝑡))
90 r19.21v 2955 . . . . . . . . . . . . . . 15 (∀𝑣𝑘 (𝑘 ⊆ (1st𝑓) → 𝑣𝑡) ↔ (𝑘 ⊆ (1st𝑓) → ∀𝑣𝑘 𝑣𝑡))
9189, 90bitr4i 267 . . . . . . . . . . . . . 14 ((𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∀𝑣𝑘 (𝑘 ⊆ (1st𝑓) → 𝑣𝑡))
9291ralbii 2975 . . . . . . . . . . . . 13 (∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∀𝑘𝐹𝑣𝑘 (𝑘 ⊆ (1st𝑓) → 𝑣𝑡))
9381, 87, 923bitr4i 292 . . . . . . . . . . . 12 (∀𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡))
9493rexbii 3035 . . . . . . . . . . 11 (∃𝑓𝐻𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∃𝑓𝐻𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡))
951rexeqi 3135 . . . . . . . . . . 11 (∃𝑓𝐻𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∃𝑓 𝑛𝐹 ({𝑛} × 𝑛)∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡))
96 vex 3192 . . . . . . . . . . . . . . . 16 𝑛 ∈ V
97 vex 3192 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
9896, 97op1std 7130 . . . . . . . . . . . . . . 15 (𝑓 = ⟨𝑛, 𝑚⟩ → (1st𝑓) = 𝑛)
9998sseq2d 3617 . . . . . . . . . . . . . 14 (𝑓 = ⟨𝑛, 𝑚⟩ → (𝑘 ⊆ (1st𝑓) ↔ 𝑘𝑛))
10099imbi1d 331 . . . . . . . . . . . . 13 (𝑓 = ⟨𝑛, 𝑚⟩ → ((𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ (𝑘𝑛𝑘𝑡)))
101100ralbidv 2981 . . . . . . . . . . . 12 (𝑓 = ⟨𝑛, 𝑚⟩ → (∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∀𝑘𝐹 (𝑘𝑛𝑘𝑡)))
102101rexiunxp 5227 . . . . . . . . . . 11 (∃𝑓 𝑛𝐹 ({𝑛} × 𝑛)∀𝑘𝐹 (𝑘 ⊆ (1st𝑓) → 𝑘𝑡) ↔ ∃𝑛𝐹𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡))
10394, 95, 1023bitri 286 . . . . . . . . . 10 (∃𝑓𝐻𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∃𝑛𝐹𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡))
104 fileln0 21573 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → 𝑛 ≠ ∅)
105104adantlr 750 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → 𝑛 ≠ ∅)
106 r19.9rzv 4042 . . . . . . . . . . . . 13 (𝑛 ≠ ∅ → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ ∃𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡)))
107105, 106syl 17 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ ∃𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡)))
108 ssid 3608 . . . . . . . . . . . . . . 15 𝑛𝑛
109 sseq1 3610 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘𝑛𝑛𝑛))
110 sseq1 3610 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘𝑡𝑛𝑡))
111109, 110imbi12d 334 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝑘𝑛𝑘𝑡) ↔ (𝑛𝑛𝑛𝑡)))
112111rspcv 3294 . . . . . . . . . . . . . . 15 (𝑛𝐹 → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) → (𝑛𝑛𝑛𝑡)))
113108, 112mpii 46 . . . . . . . . . . . . . 14 (𝑛𝐹 → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) → 𝑛𝑡))
114113adantl 482 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) → 𝑛𝑡))
115 sstr2 3594 . . . . . . . . . . . . . . 15 (𝑘𝑛 → (𝑛𝑡𝑘𝑡))
116115com12 32 . . . . . . . . . . . . . 14 (𝑛𝑡 → (𝑘𝑛𝑘𝑡))
117116ralrimivw 2962 . . . . . . . . . . . . 13 (𝑛𝑡 → ∀𝑘𝐹 (𝑘𝑛𝑘𝑡))
118114, 117impbid1 215 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → (∀𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ 𝑛𝑡))
119107, 118bitr3d 270 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) ∧ 𝑛𝐹) → (∃𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ 𝑛𝑡))
120119rexbidva 3043 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑛𝐹𝑚𝑛𝑘𝐹 (𝑘𝑛𝑘𝑡) ↔ ∃𝑛𝐹 𝑛𝑡))
121103, 120syl5bb 272 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑓𝐻𝑑𝐻 ((1st𝑑) ⊆ (1st𝑓) → (2nd𝑑) ∈ 𝑡) ↔ ∃𝑛𝐹 𝑛𝑡))
12232, 73, 1213bitrd 294 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡𝑋) → (∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡 ↔ ∃𝑛𝐹 𝑛𝑡))
123122pm5.32da 672 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → ((𝑡𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡) ↔ (𝑡𝑋 ∧ ∃𝑛𝐹 𝑛𝑡)))
124 filn0 21585 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
12596snnz 4284 . . . . . . . . . . . . . . . 16 {𝑛} ≠ ∅
126104, 125jctil 559 . . . . . . . . . . . . . . 15 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ({𝑛} ≠ ∅ ∧ 𝑛 ≠ ∅))
127 neanior 2882 . . . . . . . . . . . . . . 15 (({𝑛} ≠ ∅ ∧ 𝑛 ≠ ∅) ↔ ¬ ({𝑛} = ∅ ∨ 𝑛 = ∅))
128126, 127sylib 208 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ¬ ({𝑛} = ∅ ∨ 𝑛 = ∅))
129 ss0b 3950 . . . . . . . . . . . . . . 15 (({𝑛} × 𝑛) ⊆ ∅ ↔ ({𝑛} × 𝑛) = ∅)
130 xpeq0 5518 . . . . . . . . . . . . . . 15 (({𝑛} × 𝑛) = ∅ ↔ ({𝑛} = ∅ ∨ 𝑛 = ∅))
131129, 130bitri 264 . . . . . . . . . . . . . 14 (({𝑛} × 𝑛) ⊆ ∅ ↔ ({𝑛} = ∅ ∨ 𝑛 = ∅))
132128, 131sylnibr 319 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ¬ ({𝑛} × 𝑛) ⊆ ∅)
133132ralrimiva 2961 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → ∀𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅)
134 r19.2z 4037 . . . . . . . . . . . 12 ((𝐹 ≠ ∅ ∧ ∀𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅) → ∃𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅)
135124, 133, 134syl2anc 692 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → ∃𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅)
136 rexnal 2990 . . . . . . . . . . 11 (∃𝑛𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅ ↔ ¬ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
137135, 136sylib 208 . . . . . . . . . 10 (𝐹 ∈ (Fil‘𝑋) → ¬ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
1381sseq1i 3613 . . . . . . . . . . . 12 (𝐻 ⊆ ∅ ↔ 𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
139 ss0b 3950 . . . . . . . . . . . 12 (𝐻 ⊆ ∅ ↔ 𝐻 = ∅)
140 iunss 4532 . . . . . . . . . . . 12 ( 𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅ ↔ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
141138, 139, 1403bitr3i 290 . . . . . . . . . . 11 (𝐻 = ∅ ↔ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
142141necon3abii 2836 . . . . . . . . . 10 (𝐻 ≠ ∅ ↔ ¬ ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ∅)
143137, 142sylibr 224 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ≠ ∅)
144 dmresi 5421 . . . . . . . . . . . 12 dom ( I ↾ 𝐻) = 𝐻
1451, 2filnetlem2 32043 . . . . . . . . . . . . . 14 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
146145simpli 474 . . . . . . . . . . . . 13 ( I ↾ 𝐻) ⊆ 𝐷
147 dmss 5288 . . . . . . . . . . . . 13 (( I ↾ 𝐻) ⊆ 𝐷 → dom ( I ↾ 𝐻) ⊆ dom 𝐷)
148146, 147ax-mp 5 . . . . . . . . . . . 12 dom ( I ↾ 𝐻) ⊆ dom 𝐷
149144, 148eqsstr3i 3620 . . . . . . . . . . 11 𝐻 ⊆ dom 𝐷
150145simpri 478 . . . . . . . . . . . . 13 𝐷 ⊆ (𝐻 × 𝐻)
151 dmss 5288 . . . . . . . . . . . . 13 (𝐷 ⊆ (𝐻 × 𝐻) → dom 𝐷 ⊆ dom (𝐻 × 𝐻))
152150, 151ax-mp 5 . . . . . . . . . . . 12 dom 𝐷 ⊆ dom (𝐻 × 𝐻)
153 dmxpid 5310 . . . . . . . . . . . 12 dom (𝐻 × 𝐻) = 𝐻
154152, 153sseqtri 3621 . . . . . . . . . . 11 dom 𝐷𝐻
155149, 154eqssi 3603 . . . . . . . . . 10 𝐻 = dom 𝐷
156155tailfb 32041 . . . . . . . . 9 ((𝐷 ∈ DirRel ∧ 𝐻 ≠ ∅) → ran (tail‘𝐷) ∈ (fBas‘𝐻))
1575, 143, 156syl2anc 692 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → ran (tail‘𝐷) ∈ (fBas‘𝐻))
158 elfm 21670 . . . . . . . 8 ((𝑋𝐹 ∧ ran (tail‘𝐷) ∈ (fBas‘𝐻) ∧ (2nd𝐻):𝐻𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)) ↔ (𝑡𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡)))
15910, 157, 9, 158syl3anc 1323 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)) ↔ (𝑡𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd𝐻) “ 𝑑) ⊆ 𝑡)))
160 filfbas 21571 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
161 elfg 21594 . . . . . . . 8 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑛𝐹 𝑛𝑡)))
162160, 161syl 17 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑛𝐹 𝑛𝑡)))
163123, 159, 1623bitr4d 300 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)) ↔ 𝑡 ∈ (𝑋filGen𝐹)))
164163eqrdv 2619 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)) = (𝑋filGen𝐹))
165 fgfil 21598 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
166164, 165eqtr2d 2656 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)))
16721, 166jca 554 . . 3 (𝐹 ∈ (Fil‘𝑋) → ((2nd𝐻):dom 𝐷𝑋𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷))))
168 feq1 5988 . . . . 5 (𝑓 = (2nd𝐻) → (𝑓:dom 𝐷𝑋 ↔ (2nd𝐻):dom 𝐷𝑋))
169 oveq2 6618 . . . . . . 7 (𝑓 = (2nd𝐻) → (𝑋 FilMap 𝑓) = (𝑋 FilMap (2nd𝐻)))
170169fveq1d 6155 . . . . . 6 (𝑓 = (2nd𝐻) → ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)) = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)))
171170eqeq2d 2631 . . . . 5 (𝑓 = (2nd𝐻) → (𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)) ↔ 𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷))))
172168, 171anbi12d 746 . . . 4 (𝑓 = (2nd𝐻) → ((𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))) ↔ ((2nd𝐻):dom 𝐷𝑋𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷)))))
173172spcegv 3283 . . 3 ((2nd𝐻) ∈ V → (((2nd𝐻):dom 𝐷𝑋𝐹 = ((𝑋 FilMap (2nd𝐻))‘ran (tail‘𝐷))) → ∃𝑓(𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))))
17415, 167, 173sylc 65 . 2 (𝐹 ∈ (Fil‘𝑋) → ∃𝑓(𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))))
175 dmeq 5289 . . . . . 6 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
176175feq2d 5993 . . . . 5 (𝑑 = 𝐷 → (𝑓:dom 𝑑𝑋𝑓:dom 𝐷𝑋))
177 fveq2 6153 . . . . . . . 8 (𝑑 = 𝐷 → (tail‘𝑑) = (tail‘𝐷))
178177rneqd 5318 . . . . . . 7 (𝑑 = 𝐷 → ran (tail‘𝑑) = ran (tail‘𝐷))
179178fveq2d 6157 . . . . . 6 (𝑑 = 𝐷 → ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)) = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))
180179eqeq2d 2631 . . . . 5 (𝑑 = 𝐷 → (𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)) ↔ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))))
181176, 180anbi12d 746 . . . 4 (𝑑 = 𝐷 → ((𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))) ↔ (𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))))
182181exbidv 1847 . . 3 (𝑑 = 𝐷 → (∃𝑓(𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))) ↔ ∃𝑓(𝑓:dom 𝐷𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))))
183182rspcev 3298 . 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))))
1845, 174, 183syl2anc 692 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 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3189  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153  cop 4159   cuni 4407   ciun 4490   class class class wbr 4618  {copab 4677   I cid 4989   × cxp 5077  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  Fun wfun 5846   Fn wfn 5847  wf 5848  ontowfo 5850  cfv 5852  (class class class)co 6610  1st c1st 7118  2nd c2nd 7119  DirRelcdir 17156  tailctail 17157  fBascfbas 19662  filGencfg 19663  Filcfil 21568   FilMap cfm 21656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-dir 17158  df-tail 17159  df-fbas 19671  df-fg 19672  df-fil 21569  df-fm 21661
This theorem is referenced by:  filnet  32046
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