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Theorem filnetlem3 32702
Description: Lemma for filnet 32704. (Contributed by Jeff Hankins, 13-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
filnetlem3 (𝐻 = 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))
Distinct variable groups:   𝑥,𝑦,𝑛,𝐹   𝑥,𝐻,𝑦   𝑛,𝑋
Allowed substitution hints:   𝐷(𝑥,𝑦,𝑛)   𝐻(𝑛)   𝑋(𝑥,𝑦)

Proof of Theorem filnetlem3
Dummy variables 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmresi 5615 . . . . . 6 dom ( I ↾ 𝐻) = 𝐻
2 filnet.h . . . . . . . . 9 𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)
3 filnet.d . . . . . . . . 9 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
42, 3filnetlem2 32701 . . . . . . . 8 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
54simpli 476 . . . . . . 7 ( I ↾ 𝐻) ⊆ 𝐷
6 dmss 5478 . . . . . . 7 (( I ↾ 𝐻) ⊆ 𝐷 → dom ( I ↾ 𝐻) ⊆ dom 𝐷)
75, 6ax-mp 5 . . . . . 6 dom ( I ↾ 𝐻) ⊆ dom 𝐷
81, 7eqsstr3i 3777 . . . . 5 𝐻 ⊆ dom 𝐷
9 ssun1 3919 . . . . 5 dom 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
108, 9sstri 3753 . . . 4 𝐻 ⊆ (dom 𝐷 ∪ ran 𝐷)
11 dmrnssfld 5539 . . . 4 (dom 𝐷 ∪ ran 𝐷) ⊆ 𝐷
1210, 11sstri 3753 . . 3 𝐻 𝐷
134simpri 481 . . . . 5 𝐷 ⊆ (𝐻 × 𝐻)
14 uniss 4610 . . . . 5 (𝐷 ⊆ (𝐻 × 𝐻) → 𝐷 (𝐻 × 𝐻))
15 uniss 4610 . . . . 5 ( 𝐷 (𝐻 × 𝐻) → 𝐷 (𝐻 × 𝐻))
1613, 14, 15mp2b 10 . . . 4 𝐷 (𝐻 × 𝐻)
17 unixpss 5390 . . . . 5 (𝐻 × 𝐻) ⊆ (𝐻𝐻)
18 unidm 3899 . . . . 5 (𝐻𝐻) = 𝐻
1917, 18sseqtri 3778 . . . 4 (𝐻 × 𝐻) ⊆ 𝐻
2016, 19sstri 3753 . . 3 𝐷𝐻
2112, 20eqssi 3760 . 2 𝐻 = 𝐷
22 filelss 21877 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → 𝑛𝑋)
23 xpss2 5285 . . . . . . . 8 (𝑛𝑋 → ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
2422, 23syl 17 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
2524ralrimiva 3104 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
26 ss2iun 4688 . . . . . 6 (∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋) → 𝑛𝐹 ({𝑛} × 𝑛) ⊆ 𝑛𝐹 ({𝑛} × 𝑋))
2725, 26syl 17 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝑛𝐹 ({𝑛} × 𝑛) ⊆ 𝑛𝐹 ({𝑛} × 𝑋))
28 iunxpconst 5332 . . . . 5 𝑛𝐹 ({𝑛} × 𝑋) = (𝐹 × 𝑋)
2927, 28syl6sseq 3792 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝑛𝐹 ({𝑛} × 𝑛) ⊆ (𝐹 × 𝑋))
302, 29syl5eqss 3790 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ⊆ (𝐹 × 𝑋))
315a1i 11 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ( I ↾ 𝐻) ⊆ 𝐷)
323relopabi 5401 . . . . 5 Rel 𝐷
3331, 32jctil 561 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷))
34 simpl 474 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝐹 ∈ (Fil‘𝑋))
3530adantr 472 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝐻 ⊆ (𝐹 × 𝑋))
36 simprl 811 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑣𝐻)
3735, 36sseldd 3745 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑣 ∈ (𝐹 × 𝑋))
38 xp1st 7366 . . . . . . . . . . 11 (𝑣 ∈ (𝐹 × 𝑋) → (1st𝑣) ∈ 𝐹)
3937, 38syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → (1st𝑣) ∈ 𝐹)
40 simprr 813 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑧𝐻)
4135, 40sseldd 3745 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑧 ∈ (𝐹 × 𝑋))
42 xp1st 7366 . . . . . . . . . . 11 (𝑧 ∈ (𝐹 × 𝑋) → (1st𝑧) ∈ 𝐹)
4341, 42syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → (1st𝑧) ∈ 𝐹)
44 filinn0 21885 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (1st𝑣) ∈ 𝐹 ∧ (1st𝑧) ∈ 𝐹) → ((1st𝑣) ∩ (1st𝑧)) ≠ ∅)
4534, 39, 43, 44syl3anc 1477 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ((1st𝑣) ∩ (1st𝑧)) ≠ ∅)
46 n0 4074 . . . . . . . . 9 (((1st𝑣) ∩ (1st𝑧)) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧)))
4745, 46sylib 208 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ∃𝑢 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧)))
4836adantr 472 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑣𝐻)
49 filin 21879 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Fil‘𝑋) ∧ (1st𝑣) ∈ 𝐹 ∧ (1st𝑧) ∈ 𝐹) → ((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹)
5034, 39, 43, 49syl3anc 1477 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹)
5150adantr 472 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹)
52 simpr 479 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧)))
53 id 22 . . . . . . . . . . . . 13 (𝑛 = ((1st𝑣) ∩ (1st𝑧)) → 𝑛 = ((1st𝑣) ∩ (1st𝑧)))
5453opeliunxp2 5416 . . . . . . . . . . . 12 (⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝑛𝐹 ({𝑛} × 𝑛) ↔ (((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))))
5551, 52, 54sylanbrc 701 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝑛𝐹 ({𝑛} × 𝑛))
5655, 2syl6eleqr 2850 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻)
57 fvex 6363 . . . . . . . . . . . . . 14 (1st𝑣) ∈ V
5857inex1 4951 . . . . . . . . . . . . 13 ((1st𝑣) ∩ (1st𝑧)) ∈ V
59 vex 3343 . . . . . . . . . . . . 13 𝑢 ∈ V
6058, 59op1st 7342 . . . . . . . . . . . 12 (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) = ((1st𝑣) ∩ (1st𝑧))
61 inss1 3976 . . . . . . . . . . . 12 ((1st𝑣) ∩ (1st𝑧)) ⊆ (1st𝑣)
6260, 61eqsstri 3776 . . . . . . . . . . 11 (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑣)
63 vex 3343 . . . . . . . . . . . 12 𝑣 ∈ V
64 opex 5081 . . . . . . . . . . . 12 ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ V
652, 3, 63, 64filnetlem1 32700 . . . . . . . . . . 11 (𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ ((𝑣𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻) ∧ (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑣)))
6662, 65mpbiran2 992 . . . . . . . . . 10 (𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ (𝑣𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻))
6748, 56, 66sylanbrc 701 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩)
6840adantr 472 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑧𝐻)
69 inss2 3977 . . . . . . . . . . . 12 ((1st𝑣) ∩ (1st𝑧)) ⊆ (1st𝑧)
7060, 69eqsstri 3776 . . . . . . . . . . 11 (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑧)
71 vex 3343 . . . . . . . . . . . 12 𝑧 ∈ V
722, 3, 71, 64filnetlem1 32700 . . . . . . . . . . 11 (𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ ((𝑧𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻) ∧ (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑧)))
7370, 72mpbiran2 992 . . . . . . . . . 10 (𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ (𝑧𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻))
7468, 56, 73sylanbrc 701 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩)
75 breq2 4808 . . . . . . . . . . 11 (𝑤 = ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ → (𝑣𝐷𝑤𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩))
76 breq2 4808 . . . . . . . . . . 11 (𝑤 = ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ → (𝑧𝐷𝑤𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩))
7775, 76anbi12d 749 . . . . . . . . . 10 (𝑤 = ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ → ((𝑣𝐷𝑤𝑧𝐷𝑤) ↔ (𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∧ 𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩)))
7864, 77spcev 3440 . . . . . . . . 9 ((𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∧ 𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) → ∃𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
7967, 74, 78syl2anc 696 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ∃𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
8047, 79exlimddv 2012 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ∃𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
8180ralrimivva 3109 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → ∀𝑣𝐻𝑧𝐻𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
82 codir 5674 . . . . . 6 ((𝐻 × 𝐻) ⊆ (𝐷𝐷) ↔ ∀𝑣𝐻𝑧𝐻𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
8381, 82sylibr 224 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (𝐻 × 𝐻) ⊆ (𝐷𝐷))
84 vex 3343 . . . . . . . . . . . . 13 𝑤 ∈ V
852, 3, 63, 84filnetlem1 32700 . . . . . . . . . . . 12 (𝑣𝐷𝑤 ↔ ((𝑣𝐻𝑤𝐻) ∧ (1st𝑤) ⊆ (1st𝑣)))
8685simplbi 478 . . . . . . . . . . 11 (𝑣𝐷𝑤 → (𝑣𝐻𝑤𝐻))
8786simpld 477 . . . . . . . . . 10 (𝑣𝐷𝑤𝑣𝐻)
882, 3, 84, 71filnetlem1 32700 . . . . . . . . . . . 12 (𝑤𝐷𝑧 ↔ ((𝑤𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑤)))
8988simplbi 478 . . . . . . . . . . 11 (𝑤𝐷𝑧 → (𝑤𝐻𝑧𝐻))
9089simprd 482 . . . . . . . . . 10 (𝑤𝐷𝑧𝑧𝐻)
9187, 90anim12i 591 . . . . . . . . 9 ((𝑣𝐷𝑤𝑤𝐷𝑧) → (𝑣𝐻𝑧𝐻))
9288simprbi 483 . . . . . . . . . 10 (𝑤𝐷𝑧 → (1st𝑧) ⊆ (1st𝑤))
9385simprbi 483 . . . . . . . . . 10 (𝑣𝐷𝑤 → (1st𝑤) ⊆ (1st𝑣))
9492, 93sylan9ssr 3758 . . . . . . . . 9 ((𝑣𝐷𝑤𝑤𝐷𝑧) → (1st𝑧) ⊆ (1st𝑣))
952, 3, 63, 71filnetlem1 32700 . . . . . . . . 9 (𝑣𝐷𝑧 ↔ ((𝑣𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑣)))
9691, 94, 95sylanbrc 701 . . . . . . . 8 ((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧)
9796ax-gen 1871 . . . . . . 7 𝑧((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧)
9897gen2 1872 . . . . . 6 𝑣𝑤𝑧((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧)
99 cotr 5666 . . . . . 6 ((𝐷𝐷) ⊆ 𝐷 ↔ ∀𝑣𝑤𝑧((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧))
10098, 99mpbir 221 . . . . 5 (𝐷𝐷) ⊆ 𝐷
10183, 100jctil 561 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ((𝐷𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (𝐷𝐷)))
102 filtop 21880 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
103 xpexg 7126 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑋𝐹) → (𝐹 × 𝑋) ∈ V)
104102, 103mpdan 705 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → (𝐹 × 𝑋) ∈ V)
105104, 30ssexd 4957 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ∈ V)
106 xpexg 7126 . . . . . . 7 ((𝐻 ∈ V ∧ 𝐻 ∈ V) → (𝐻 × 𝐻) ∈ V)
107105, 105, 106syl2anc 696 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐻 × 𝐻) ∈ V)
108 ssexg 4956 . . . . . 6 ((𝐷 ⊆ (𝐻 × 𝐻) ∧ (𝐻 × 𝐻) ∈ V) → 𝐷 ∈ V)
10913, 107, 108sylancr 698 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ V)
11021isdir 17453 . . . . 5 (𝐷 ∈ V → (𝐷 ∈ DirRel ↔ ((Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷) ∧ ((𝐷𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (𝐷𝐷)))))
111109, 110syl 17 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝐷 ∈ DirRel ↔ ((Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷) ∧ ((𝐷𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (𝐷𝐷)))))
11233, 101, 111mpbir2and 995 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ DirRel)
11330, 112jca 555 . 2 (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel))
11421, 113pm3.2i 470 1 (𝐻 = 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wex 1853  wcel 2139  wne 2932  wral 3050  Vcvv 3340  cun 3713  cin 3714  wss 3715  c0 4058  {csn 4321  cop 4327   cuni 4588   ciun 4672   class class class wbr 4804  {copab 4864   I cid 5173   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267  cres 5268  ccom 5270  Rel wrel 5271  cfv 6049  1st c1st 7332  DirRelcdir 17449  Filcfil 21870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-1st 7334  df-dir 17451  df-fbas 19965  df-fil 21871
This theorem is referenced by:  filnetlem4  32703
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