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Theorem filnetlem3 36381
Description: Lemma for filnet 36383. (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 6070 . . . . . 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 36380 . . . . . . . 8 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
54simpli 483 . . . . . . 7 ( I ↾ 𝐻) ⊆ 𝐷
6 dmss 5913 . . . . . . 7 (( I ↾ 𝐻) ⊆ 𝐷 → dom ( I ↾ 𝐻) ⊆ dom 𝐷)
75, 6ax-mp 5 . . . . . 6 dom ( I ↾ 𝐻) ⊆ dom 𝐷
81, 7eqsstrri 4031 . . . . 5 𝐻 ⊆ dom 𝐷
9 ssun1 4178 . . . . 5 dom 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
108, 9sstri 3993 . . . 4 𝐻 ⊆ (dom 𝐷 ∪ ran 𝐷)
11 dmrnssfld 5984 . . . 4 (dom 𝐷 ∪ ran 𝐷) ⊆ 𝐷
1210, 11sstri 3993 . . 3 𝐻 𝐷
134simpri 485 . . . . 5 𝐷 ⊆ (𝐻 × 𝐻)
14 uniss 4915 . . . . 5 (𝐷 ⊆ (𝐻 × 𝐻) → 𝐷 (𝐻 × 𝐻))
15 uniss 4915 . . . . 5 ( 𝐷 (𝐻 × 𝐻) → 𝐷 (𝐻 × 𝐻))
1613, 14, 15mp2b 10 . . . 4 𝐷 (𝐻 × 𝐻)
17 unixpss 5820 . . . . 5 (𝐻 × 𝐻) ⊆ (𝐻𝐻)
18 unidm 4157 . . . . 5 (𝐻𝐻) = 𝐻
1917, 18sseqtri 4032 . . . 4 (𝐻 × 𝐻) ⊆ 𝐻
2016, 19sstri 3993 . . 3 𝐷𝐻
2112, 20eqssi 4000 . 2 𝐻 = 𝐷
22 filelss 23860 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → 𝑛𝑋)
23 xpss2 5705 . . . . . . . 8 (𝑛𝑋 → ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
2422, 23syl 17 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
2524ralrimiva 3146 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
26 ss2iun 5010 . . . . . 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 5758 . . . . 5 𝑛𝐹 ({𝑛} × 𝑋) = (𝐹 × 𝑋)
2927, 28sseqtrdi 4024 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝑛𝐹 ({𝑛} × 𝑛) ⊆ (𝐹 × 𝑋))
302, 29eqsstrid 4022 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ⊆ (𝐹 × 𝑋))
315a1i 11 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ( I ↾ 𝐻) ⊆ 𝐷)
323relopabiv 5830 . . . . 5 Rel 𝐷
3331, 32jctil 519 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷))
34 simpl 482 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝐹 ∈ (Fil‘𝑋))
3530adantr 480 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝐻 ⊆ (𝐹 × 𝑋))
36 simprl 771 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑣𝐻)
3735, 36sseldd 3984 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑣 ∈ (𝐹 × 𝑋))
38 xp1st 8046 . . . . . . . . . . 11 (𝑣 ∈ (𝐹 × 𝑋) → (1st𝑣) ∈ 𝐹)
3937, 38syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → (1st𝑣) ∈ 𝐹)
40 simprr 773 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑧𝐻)
4135, 40sseldd 3984 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑧 ∈ (𝐹 × 𝑋))
42 xp1st 8046 . . . . . . . . . . 11 (𝑧 ∈ (𝐹 × 𝑋) → (1st𝑧) ∈ 𝐹)
4341, 42syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → (1st𝑧) ∈ 𝐹)
44 filinn0 23868 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (1st𝑣) ∈ 𝐹 ∧ (1st𝑧) ∈ 𝐹) → ((1st𝑣) ∩ (1st𝑧)) ≠ ∅)
4534, 39, 43, 44syl3anc 1373 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ((1st𝑣) ∩ (1st𝑧)) ≠ ∅)
46 n0 4353 . . . . . . . . 9 (((1st𝑣) ∩ (1st𝑧)) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧)))
4745, 46sylib 218 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ∃𝑢 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧)))
4836adantr 480 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑣𝐻)
49 filin 23862 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Fil‘𝑋) ∧ (1st𝑣) ∈ 𝐹 ∧ (1st𝑧) ∈ 𝐹) → ((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹)
5034, 39, 43, 49syl3anc 1373 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹)
5150adantr 480 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹)
52 simpr 484 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧)))
53 id 22 . . . . . . . . . . . . 13 (𝑛 = ((1st𝑣) ∩ (1st𝑧)) → 𝑛 = ((1st𝑣) ∩ (1st𝑧)))
5453opeliunxp2 5849 . . . . . . . . . . . 12 (⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝑛𝐹 ({𝑛} × 𝑛) ↔ (((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))))
5551, 52, 54sylanbrc 583 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝑛𝐹 ({𝑛} × 𝑛))
5655, 2eleqtrrdi 2852 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻)
57 fvex 6919 . . . . . . . . . . . . . 14 (1st𝑣) ∈ V
5857inex1 5317 . . . . . . . . . . . . 13 ((1st𝑣) ∩ (1st𝑧)) ∈ V
59 vex 3484 . . . . . . . . . . . . 13 𝑢 ∈ V
6058, 59op1st 8022 . . . . . . . . . . . 12 (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) = ((1st𝑣) ∩ (1st𝑧))
61 inss1 4237 . . . . . . . . . . . 12 ((1st𝑣) ∩ (1st𝑧)) ⊆ (1st𝑣)
6260, 61eqsstri 4030 . . . . . . . . . . 11 (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑣)
63 vex 3484 . . . . . . . . . . . 12 𝑣 ∈ V
64 opex 5469 . . . . . . . . . . . 12 ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ V
652, 3, 63, 64filnetlem1 36379 . . . . . . . . . . 11 (𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ ((𝑣𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻) ∧ (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑣)))
6662, 65mpbiran2 710 . . . . . . . . . 10 (𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ (𝑣𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻))
6748, 56, 66sylanbrc 583 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩)
6840adantr 480 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑧𝐻)
69 inss2 4238 . . . . . . . . . . . 12 ((1st𝑣) ∩ (1st𝑧)) ⊆ (1st𝑧)
7060, 69eqsstri 4030 . . . . . . . . . . 11 (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑧)
71 vex 3484 . . . . . . . . . . . 12 𝑧 ∈ V
722, 3, 71, 64filnetlem1 36379 . . . . . . . . . . 11 (𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ ((𝑧𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻) ∧ (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑧)))
7370, 72mpbiran2 710 . . . . . . . . . 10 (𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ (𝑧𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻))
7468, 56, 73sylanbrc 583 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩)
75 breq2 5147 . . . . . . . . . . 11 (𝑤 = ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ → (𝑣𝐷𝑤𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩))
76 breq2 5147 . . . . . . . . . . 11 (𝑤 = ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ → (𝑧𝐷𝑤𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩))
7775, 76anbi12d 632 . . . . . . . . . 10 (𝑤 = ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ → ((𝑣𝐷𝑤𝑧𝐷𝑤) ↔ (𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∧ 𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩)))
7864, 77spcev 3606 . . . . . . . . 9 ((𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∧ 𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) → ∃𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
7967, 74, 78syl2anc 584 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ∃𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
8047, 79exlimddv 1935 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ∃𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
8180ralrimivva 3202 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → ∀𝑣𝐻𝑧𝐻𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
82 codir 6140 . . . . . 6 ((𝐻 × 𝐻) ⊆ (𝐷𝐷) ↔ ∀𝑣𝐻𝑧𝐻𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
8381, 82sylibr 234 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (𝐻 × 𝐻) ⊆ (𝐷𝐷))
84 vex 3484 . . . . . . . . . . . . 13 𝑤 ∈ V
852, 3, 63, 84filnetlem1 36379 . . . . . . . . . . . 12 (𝑣𝐷𝑤 ↔ ((𝑣𝐻𝑤𝐻) ∧ (1st𝑤) ⊆ (1st𝑣)))
8685simplbi 497 . . . . . . . . . . 11 (𝑣𝐷𝑤 → (𝑣𝐻𝑤𝐻))
8786simpld 494 . . . . . . . . . 10 (𝑣𝐷𝑤𝑣𝐻)
882, 3, 84, 71filnetlem1 36379 . . . . . . . . . . . 12 (𝑤𝐷𝑧 ↔ ((𝑤𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑤)))
8988simplbi 497 . . . . . . . . . . 11 (𝑤𝐷𝑧 → (𝑤𝐻𝑧𝐻))
9089simprd 495 . . . . . . . . . 10 (𝑤𝐷𝑧𝑧𝐻)
9187, 90anim12i 613 . . . . . . . . 9 ((𝑣𝐷𝑤𝑤𝐷𝑧) → (𝑣𝐻𝑧𝐻))
9288simprbi 496 . . . . . . . . . 10 (𝑤𝐷𝑧 → (1st𝑧) ⊆ (1st𝑤))
9385simprbi 496 . . . . . . . . . 10 (𝑣𝐷𝑤 → (1st𝑤) ⊆ (1st𝑣))
9492, 93sylan9ssr 3998 . . . . . . . . 9 ((𝑣𝐷𝑤𝑤𝐷𝑧) → (1st𝑧) ⊆ (1st𝑣))
952, 3, 63, 71filnetlem1 36379 . . . . . . . . 9 (𝑣𝐷𝑧 ↔ ((𝑣𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑣)))
9691, 94, 95sylanbrc 583 . . . . . . . 8 ((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧)
9796ax-gen 1795 . . . . . . 7 𝑧((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧)
9897gen2 1796 . . . . . 6 𝑣𝑤𝑧((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧)
99 cotr 6130 . . . . . 6 ((𝐷𝐷) ⊆ 𝐷 ↔ ∀𝑣𝑤𝑧((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧))
10098, 99mpbir 231 . . . . 5 (𝐷𝐷) ⊆ 𝐷
10183, 100jctil 519 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ((𝐷𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (𝐷𝐷)))
102 filtop 23863 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
103 xpexg 7770 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑋𝐹) → (𝐹 × 𝑋) ∈ V)
104102, 103mpdan 687 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → (𝐹 × 𝑋) ∈ V)
105104, 30ssexd 5324 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ∈ V)
106105, 105xpexd 7771 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐻 × 𝐻) ∈ V)
107 ssexg 5323 . . . . . 6 ((𝐷 ⊆ (𝐻 × 𝐻) ∧ (𝐻 × 𝐻) ∈ V) → 𝐷 ∈ V)
10813, 106, 107sylancr 587 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ V)
10921isdir 18643 . . . . 5 (𝐷 ∈ V → (𝐷 ∈ DirRel ↔ ((Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷) ∧ ((𝐷𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (𝐷𝐷)))))
110108, 109syl 17 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝐷 ∈ DirRel ↔ ((Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷) ∧ ((𝐷𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (𝐷𝐷)))))
11133, 101, 110mpbir2and 713 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ DirRel)
11230, 111jca 511 . 2 (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel))
11321, 112pm3.2i 470 1 (𝐻 = 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  Vcvv 3480  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626  cop 4632   cuni 4907   ciun 4991   class class class wbr 5143  {copab 5205   I cid 5577   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  ccom 5689  Rel wrel 5690  cfv 6561  1st c1st 8012  DirRelcdir 18639  Filcfil 23853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-dir 18641  df-fbas 21361  df-fil 23854
This theorem is referenced by:  filnetlem4  36382
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