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Theorem filnetlem3 34784
Description: Lemma for filnet 34786. (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 6003 . . . . . 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 34783 . . . . . . . 8 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
54simpli 484 . . . . . . 7 ( I ↾ 𝐻) ⊆ 𝐷
6 dmss 5856 . . . . . . 7 (( I ↾ 𝐻) ⊆ 𝐷 → dom ( I ↾ 𝐻) ⊆ dom 𝐷)
75, 6ax-mp 5 . . . . . 6 dom ( I ↾ 𝐻) ⊆ dom 𝐷
81, 7eqsstrri 3977 . . . . 5 𝐻 ⊆ dom 𝐷
9 ssun1 4130 . . . . 5 dom 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
108, 9sstri 3951 . . . 4 𝐻 ⊆ (dom 𝐷 ∪ ran 𝐷)
11 dmrnssfld 5923 . . . 4 (dom 𝐷 ∪ ran 𝐷) ⊆ 𝐷
1210, 11sstri 3951 . . 3 𝐻 𝐷
134simpri 486 . . . . 5 𝐷 ⊆ (𝐻 × 𝐻)
14 uniss 4871 . . . . 5 (𝐷 ⊆ (𝐻 × 𝐻) → 𝐷 (𝐻 × 𝐻))
15 uniss 4871 . . . . 5 ( 𝐷 (𝐻 × 𝐻) → 𝐷 (𝐻 × 𝐻))
1613, 14, 15mp2b 10 . . . 4 𝐷 (𝐻 × 𝐻)
17 unixpss 5764 . . . . 5 (𝐻 × 𝐻) ⊆ (𝐻𝐻)
18 unidm 4110 . . . . 5 (𝐻𝐻) = 𝐻
1917, 18sseqtri 3978 . . . 4 (𝐻 × 𝐻) ⊆ 𝐻
2016, 19sstri 3951 . . 3 𝐷𝐻
2112, 20eqssi 3958 . 2 𝐻 = 𝐷
22 filelss 23149 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → 𝑛𝑋)
23 xpss2 5651 . . . . . . . 8 (𝑛𝑋 → ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
2422, 23syl 17 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛𝐹) → ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
2524ralrimiva 3141 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → ∀𝑛𝐹 ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
26 ss2iun 4970 . . . . . 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 5702 . . . . 5 𝑛𝐹 ({𝑛} × 𝑋) = (𝐹 × 𝑋)
2927, 28sseqtrdi 3992 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝑛𝐹 ({𝑛} × 𝑛) ⊆ (𝐹 × 𝑋))
302, 29eqsstrid 3990 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ⊆ (𝐹 × 𝑋))
315a1i 11 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ( I ↾ 𝐻) ⊆ 𝐷)
323relopabiv 5774 . . . . 5 Rel 𝐷
3331, 32jctil 520 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷))
34 simpl 483 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝐹 ∈ (Fil‘𝑋))
3530adantr 481 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝐻 ⊆ (𝐹 × 𝑋))
36 simprl 769 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑣𝐻)
3735, 36sseldd 3943 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑣 ∈ (𝐹 × 𝑋))
38 xp1st 7945 . . . . . . . . . . 11 (𝑣 ∈ (𝐹 × 𝑋) → (1st𝑣) ∈ 𝐹)
3937, 38syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → (1st𝑣) ∈ 𝐹)
40 simprr 771 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑧𝐻)
4135, 40sseldd 3943 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → 𝑧 ∈ (𝐹 × 𝑋))
42 xp1st 7945 . . . . . . . . . . 11 (𝑧 ∈ (𝐹 × 𝑋) → (1st𝑧) ∈ 𝐹)
4341, 42syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → (1st𝑧) ∈ 𝐹)
44 filinn0 23157 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (1st𝑣) ∈ 𝐹 ∧ (1st𝑧) ∈ 𝐹) → ((1st𝑣) ∩ (1st𝑧)) ≠ ∅)
4534, 39, 43, 44syl3anc 1371 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ((1st𝑣) ∩ (1st𝑧)) ≠ ∅)
46 n0 4304 . . . . . . . . 9 (((1st𝑣) ∩ (1st𝑧)) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧)))
4745, 46sylib 217 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ∃𝑢 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧)))
4836adantr 481 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑣𝐻)
49 filin 23151 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Fil‘𝑋) ∧ (1st𝑣) ∈ 𝐹 ∧ (1st𝑧) ∈ 𝐹) → ((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹)
5034, 39, 43, 49syl3anc 1371 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹)
5150adantr 481 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ((1st𝑣) ∩ (1st𝑧)) ∈ 𝐹)
52 simpr 485 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧)))
53 id 22 . . . . . . . . . . . . 13 (𝑛 = ((1st𝑣) ∩ (1st𝑧)) → 𝑛 = ((1st𝑣) ∩ (1st𝑧)))
5453opeliunxp2 5792 . . . . . . . . . . . 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 2849 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻)
57 fvex 6852 . . . . . . . . . . . . . 14 (1st𝑣) ∈ V
5857inex1 5272 . . . . . . . . . . . . 13 ((1st𝑣) ∩ (1st𝑧)) ∈ V
59 vex 3447 . . . . . . . . . . . . 13 𝑢 ∈ V
6058, 59op1st 7921 . . . . . . . . . . . 12 (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) = ((1st𝑣) ∩ (1st𝑧))
61 inss1 4186 . . . . . . . . . . . 12 ((1st𝑣) ∩ (1st𝑧)) ⊆ (1st𝑣)
6260, 61eqsstri 3976 . . . . . . . . . . 11 (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑣)
63 vex 3447 . . . . . . . . . . . 12 𝑣 ∈ V
64 opex 5419 . . . . . . . . . . . 12 ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ V
652, 3, 63, 64filnetlem1 34782 . . . . . . . . . . 11 (𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ ((𝑣𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻) ∧ (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑣)))
6662, 65mpbiran2 708 . . . . . . . . . 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 481 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) ∧ 𝑢 ∈ ((1st𝑣) ∩ (1st𝑧))) → 𝑧𝐻)
69 inss2 4187 . . . . . . . . . . . 12 ((1st𝑣) ∩ (1st𝑧)) ⊆ (1st𝑧)
7060, 69eqsstri 3976 . . . . . . . . . . 11 (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑧)
71 vex 3447 . . . . . . . . . . . 12 𝑧 ∈ V
722, 3, 71, 64filnetlem1 34782 . . . . . . . . . . 11 (𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ↔ ((𝑧𝐻 ∧ ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∈ 𝐻) ∧ (1st ‘⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩) ⊆ (1st𝑧)))
7370, 72mpbiran2 708 . . . . . . . . . 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 5107 . . . . . . . . . . 11 (𝑤 = ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ → (𝑣𝐷𝑤𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩))
76 breq2 5107 . . . . . . . . . . 11 (𝑤 = ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ → (𝑧𝐷𝑤𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩))
7775, 76anbi12d 631 . . . . . . . . . 10 (𝑤 = ⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ → ((𝑣𝐷𝑤𝑧𝐷𝑤) ↔ (𝑣𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩ ∧ 𝑧𝐷⟨((1st𝑣) ∩ (1st𝑧)), 𝑢⟩)))
7864, 77spcev 3563 . . . . . . . . 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 1938 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣𝐻𝑧𝐻)) → ∃𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
8180ralrimivva 3195 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → ∀𝑣𝐻𝑧𝐻𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
82 codir 6072 . . . . . 6 ((𝐻 × 𝐻) ⊆ (𝐷𝐷) ↔ ∀𝑣𝐻𝑧𝐻𝑤(𝑣𝐷𝑤𝑧𝐷𝑤))
8381, 82sylibr 233 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (𝐻 × 𝐻) ⊆ (𝐷𝐷))
84 vex 3447 . . . . . . . . . . . . 13 𝑤 ∈ V
852, 3, 63, 84filnetlem1 34782 . . . . . . . . . . . 12 (𝑣𝐷𝑤 ↔ ((𝑣𝐻𝑤𝐻) ∧ (1st𝑤) ⊆ (1st𝑣)))
8685simplbi 498 . . . . . . . . . . 11 (𝑣𝐷𝑤 → (𝑣𝐻𝑤𝐻))
8786simpld 495 . . . . . . . . . 10 (𝑣𝐷𝑤𝑣𝐻)
882, 3, 84, 71filnetlem1 34782 . . . . . . . . . . . 12 (𝑤𝐷𝑧 ↔ ((𝑤𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑤)))
8988simplbi 498 . . . . . . . . . . 11 (𝑤𝐷𝑧 → (𝑤𝐻𝑧𝐻))
9089simprd 496 . . . . . . . . . 10 (𝑤𝐷𝑧𝑧𝐻)
9187, 90anim12i 613 . . . . . . . . 9 ((𝑣𝐷𝑤𝑤𝐷𝑧) → (𝑣𝐻𝑧𝐻))
9288simprbi 497 . . . . . . . . . 10 (𝑤𝐷𝑧 → (1st𝑧) ⊆ (1st𝑤))
9385simprbi 497 . . . . . . . . . 10 (𝑣𝐷𝑤 → (1st𝑤) ⊆ (1st𝑣))
9492, 93sylan9ssr 3956 . . . . . . . . 9 ((𝑣𝐷𝑤𝑤𝐷𝑧) → (1st𝑧) ⊆ (1st𝑣))
952, 3, 63, 71filnetlem1 34782 . . . . . . . . 9 (𝑣𝐷𝑧 ↔ ((𝑣𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑣)))
9691, 94, 95sylanbrc 583 . . . . . . . 8 ((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧)
9796ax-gen 1797 . . . . . . 7 𝑧((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧)
9897gen2 1798 . . . . . 6 𝑣𝑤𝑧((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧)
99 cotr 6062 . . . . . 6 ((𝐷𝐷) ⊆ 𝐷 ↔ ∀𝑣𝑤𝑧((𝑣𝐷𝑤𝑤𝐷𝑧) → 𝑣𝐷𝑧))
10098, 99mpbir 230 . . . . 5 (𝐷𝐷) ⊆ 𝐷
10183, 100jctil 520 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ((𝐷𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (𝐷𝐷)))
102 filtop 23152 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
103 xpexg 7676 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑋𝐹) → (𝐹 × 𝑋) ∈ V)
104102, 103mpdan 685 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → (𝐹 × 𝑋) ∈ V)
105104, 30ssexd 5279 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝐻 ∈ V)
106105, 105xpexd 7677 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐻 × 𝐻) ∈ V)
107 ssexg 5278 . . . . . 6 ((𝐷 ⊆ (𝐻 × 𝐻) ∧ (𝐻 × 𝐻) ∈ V) → 𝐷 ∈ V)
10813, 106, 107sylancr 587 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ V)
10921isdir 18441 . . . . 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 711 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ DirRel)
11230, 111jca 512 . 2 (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel))
11321, 112pm3.2i 471 1 (𝐻 = 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  wne 2941  wral 3062  Vcvv 3443  cun 3906  cin 3907  wss 3908  c0 4280  {csn 4584  cop 4590   cuni 4863   ciun 4952   class class class wbr 5103  {copab 5165   I cid 5528   × cxp 5629  ccnv 5630  dom cdm 5631  ran crn 5632  cres 5633  ccom 5635  Rel wrel 5636  cfv 6493  1st c1st 7911  DirRelcdir 18437  Filcfil 23142
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-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
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 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-1st 7913  df-dir 18439  df-fbas 20740  df-fil 23143
This theorem is referenced by:  filnetlem4  34785
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