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Theorem filnetlem3 35568
Description: Lemma for filnet 35570. (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 6050 . . . . . 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 35567 . . . . . . . 8 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
54simpli 482 . . . . . . 7 ( I ↾ 𝐻) ⊆ 𝐷
6 dmss 5901 . . . . . . 7 (( I ↾ 𝐻) ⊆ 𝐷 → dom ( I ↾ 𝐻) ⊆ dom 𝐷)
75, 6ax-mp 5 . . . . . 6 dom ( I ↾ 𝐻) ⊆ dom 𝐷
81, 7eqsstrri 4016 . . . . 5 𝐻 ⊆ dom 𝐷
9 ssun1 4171 . . . . 5 dom 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
108, 9sstri 3990 . . . 4 𝐻 ⊆ (dom 𝐷 ∪ ran 𝐷)
11 dmrnssfld 5968 . . . 4 (dom 𝐷 ∪ ran 𝐷) ⊆ 𝐷
1210, 11sstri 3990 . . 3 𝐻 𝐷
134simpri 484 . . . . 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 5809 . . . . 5 (𝐻 × 𝐻) ⊆ (𝐻𝐻)
18 unidm 4151 . . . . 5 (𝐻𝐻) = 𝐻
1917, 18sseqtri 4017 . . . 4 (𝐻 × 𝐻) ⊆ 𝐻
2016, 19sstri 3990 . . 3 𝐷𝐻
2112, 20eqssi 3997 . 2 𝐻 = 𝐷
22 filelss 23576 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → 𝑛𝑋)
23 xpss2 5695 . . . . . . . 8 (𝑛𝑋 → ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
2422, 23syl 17 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
2524ralrimiva 3144 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
26 ss2iun 5014 . . . . . 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 5747 . . . . 5 𝑛𝐹 ({𝑛} × 𝑋) = (𝐹 × 𝑋)
2927, 28sseqtrdi 4031 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝑛𝐹 ({𝑛} × 𝑛) ⊆ (𝐹 × 𝑋))
302, 29eqsstrid 4029 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ⊆ (𝐹 × 𝑋))
315a1i 11 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ( I ↾ 𝐻) ⊆ 𝐷)
323relopabiv 5819 . . . . 5 Rel 𝐷
3331, 32jctil 518 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷))
34 simpl 481 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝐹 ∈ (Fil‘𝑋))
3530adantr 479 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝐻 ⊆ (𝐹 × 𝑋))
36 simprl 767 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑣𝐻)
3735, 36sseldd 3982 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑣 ∈ (𝐹 × 𝑋))
38 xp1st 8009 . . . . . . . . . . 11 (𝑣 ∈ (𝐹 × 𝑋) → (1st𝑣) ∈ 𝐹)
3937, 38syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → (1st𝑣) ∈ 𝐹)
40 simprr 769 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑧𝐻)
4135, 40sseldd 3982 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑧 ∈ (𝐹 × 𝑋))
42 xp1st 8009 . . . . . . . . . . 11 (𝑧 ∈ (𝐹 × 𝑋) → (1st𝑧) ∈ 𝐹)
4341, 42syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → (1st𝑧) ∈ 𝐹)
44 filinn0 23584 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (1st𝑣) ∈ 𝐹 ∧ (1st𝑧) ∈ 𝐹) → ((1st𝑣) ∩ (1st𝑧)) ≠ ∅)
4534, 39, 43, 44syl3anc 1369 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ((1st𝑣) ∩ (1st𝑧)) ≠ ∅)
46 n0 4345 . . . . . . . . 9 (((1st𝑣) ∩ (1st𝑧)) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧)))
4745, 46sylib 217 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ∃𝑢 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧)))
4836adantr 479 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑣𝐻)
49 filin 23578 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Fil‘𝑋) ∧ (1st𝑣) ∈ 𝐹 ∧ (1st𝑧) ∈ 𝐹) → ((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹)
5034, 39, 43, 49syl3anc 1369 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹)
5150adantr 479 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹)
52 simpr 483 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧)))
53 id 22 . . . . . . . . . . . . 13 (𝑛 = ((1st𝑣) ∩ (1st𝑧)) → 𝑛 = ((1st𝑣) ∩ (1st𝑧)))
5453opeliunxp2 5837 . . . . . . . . . . . 12 (⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝑛𝐹 ({𝑛} × 𝑛) ↔ (((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))))
5551, 52, 54sylanbrc 581 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝑛𝐹 ({𝑛} × 𝑛))
5655, 2eleqtrrdi 2842 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻)
57 fvex 6903 . . . . . . . . . . . . . 14 (1st𝑣) ∈ V
5857inex1 5316 . . . . . . . . . . . . 13 ((1st𝑣) ∩ (1st𝑧)) ∈ V
59 vex 3476 . . . . . . . . . . . . 13 𝑢 ∈ V
6058, 59op1st 7985 . . . . . . . . . . . 12 (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) = ((1st𝑣) ∩ (1st𝑧))
61 inss1 4227 . . . . . . . . . . . 12 ((1st𝑣) ∩ (1st𝑧)) ⊆ (1st𝑣)
6260, 61eqsstri 4015 . . . . . . . . . . 11 (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑣)
63 vex 3476 . . . . . . . . . . . 12 𝑣 ∈ V
64 opex 5463 . . . . . . . . . . . 12 ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ V
652, 3, 63, 64filnetlem1 35566 . . . . . . . . . . 11 (𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ ((𝑣𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻) ∧ (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑣)))
6662, 65mpbiran2 706 . . . . . . . . . 10 (𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ (𝑣𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻))
6748, 56, 66sylanbrc 581 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩)
6840adantr 479 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑧𝐻)
69 inss2 4228 . . . . . . . . . . . 12 ((1st𝑣) ∩ (1st𝑧)) ⊆ (1st𝑧)
7060, 69eqsstri 4015 . . . . . . . . . . 11 (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑧)
71 vex 3476 . . . . . . . . . . . 12 𝑧 ∈ V
722, 3, 71, 64filnetlem1 35566 . . . . . . . . . . 11 (𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ ((𝑧𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻) ∧ (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑧)))
7370, 72mpbiran2 706 . . . . . . . . . 10 (𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ (𝑧𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻))
7468, 56, 73sylanbrc 581 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩)
75 breq2 5151 . . . . . . . . . . 11 (𝑤 = ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ → (𝑣𝐷𝑤𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩))
76 breq2 5151 . . . . . . . . . . 11 (𝑤 = ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ → (𝑧𝐷𝑤𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩))
7775, 76anbi12d 629 . . . . . . . . . 10 (𝑤 = ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ → ((𝑣𝐷𝑤𝑧𝐷𝑤) ↔ (𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∧ 𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩)))
7864, 77spcev 3595 . . . . . . . . 9 ((𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∧ 𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) → ∃𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
7967, 74, 78syl2anc 582 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ∃𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
8047, 79exlimddv 1936 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ∃𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
8180ralrimivva 3198 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → ∀𝑣𝐻𝑧𝐻𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
82 codir 6120 . . . . . 6 ((𝐻 × 𝐻) ⊆ (𝐷𝐷) ↔ ∀𝑣𝐻𝑧𝐻𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
8381, 82sylibr 233 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (𝐻 × 𝐻) ⊆ (𝐷𝐷))
84 vex 3476 . . . . . . . . . . . . 13 𝑤 ∈ V
852, 3, 63, 84filnetlem1 35566 . . . . . . . . . . . 12 (𝑣𝐷𝑤 ↔ ((𝑣𝐻𝑤𝐻) ∧ (1st𝑤) ⊆ (1st𝑣)))
8685simplbi 496 . . . . . . . . . . 11 (𝑣𝐷𝑤 → (𝑣𝐻𝑤𝐻))
8786simpld 493 . . . . . . . . . 10 (𝑣𝐷𝑤𝑣𝐻)
882, 3, 84, 71filnetlem1 35566 . . . . . . . . . . . 12 (𝑤𝐷𝑧 ↔ ((𝑤𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑤)))
8988simplbi 496 . . . . . . . . . . 11 (𝑤𝐷𝑧 → (𝑤𝐻𝑧𝐻))
9089simprd 494 . . . . . . . . . 10 (𝑤𝐷𝑧𝑧𝐻)
9187, 90anim12i 611 . . . . . . . . 9 ((𝑣𝐷𝑤𝑤𝐷𝑧) → (𝑣𝐻𝑧𝐻))
9288simprbi 495 . . . . . . . . . 10 (𝑤𝐷𝑧 → (1st𝑧) ⊆ (1st𝑤))
9385simprbi 495 . . . . . . . . . 10 (𝑣𝐷𝑤 → (1st𝑤) ⊆ (1st𝑣))
9492, 93sylan9ssr 3995 . . . . . . . . 9 ((𝑣𝐷𝑤𝑤𝐷𝑧) → (1st𝑧) ⊆ (1st𝑣))
952, 3, 63, 71filnetlem1 35566 . . . . . . . . 9 (𝑣𝐷𝑧 ↔ ((𝑣𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑣)))
9691, 94, 95sylanbrc 581 . . . . . . . 8 ((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧)
9796ax-gen 1795 . . . . . . 7 𝑧((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧)
9897gen2 1796 . . . . . 6 𝑣𝑤𝑧((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧)
99 cotr 6110 . . . . . 6 ((𝐷𝐷) ⊆ 𝐷 ↔ ∀𝑣𝑤𝑧((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧))
10098, 99mpbir 230 . . . . 5 (𝐷𝐷) ⊆ 𝐷
10183, 100jctil 518 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ((𝐷𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (𝐷𝐷)))
102 filtop 23579 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
103 xpexg 7739 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑋𝐹) → (𝐹 × 𝑋) ∈ V)
104102, 103mpdan 683 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → (𝐹 × 𝑋) ∈ V)
105104, 30ssexd 5323 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ∈ V)
106105, 105xpexd 7740 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐻 × 𝐻) ∈ V)
107 ssexg 5322 . . . . . 6 ((𝐷 ⊆ (𝐻 × 𝐻) ∧ (𝐻 × 𝐻) ∈ V) → 𝐷 ∈ V)
10813, 106, 107sylancr 585 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ V)
10921isdir 18555 . . . . 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 709 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ DirRel)
11230, 111jca 510 . 2 (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel))
11321, 112pm3.2i 469 1 (𝐻 = 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1537   = wceq 1539  wex 1779  wcel 2104  wne 2938  wral 3059  Vcvv 3472  cun 3945  cin 3946  wss 3947  c0 4321  {csn 4627  cop 4633   cuni 4907   ciun 4996   class class class wbr 5147  {copab 5209   I cid 5572   × cxp 5673  ccnv 5674  dom cdm 5675  ran crn 5676  cres 5677  ccom 5679  Rel wrel 5680  cfv 6542  1st c1st 7975  DirRelcdir 18551  Filcfil 23569
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1st 7977  df-dir 18553  df-fbas 21141  df-fil 23570
This theorem is referenced by:  filnetlem4  35569
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