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Theorem filconn 21627
Description: A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
filconn (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)

Proof of Theorem filconn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2 filunibas 21625 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
32fveq2d 6162 . . 3 (𝐹 ∈ (Fil‘𝑋) → (Fil‘ 𝐹) = (Fil‘𝑋))
41, 3eleqtrrd 2701 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (Fil‘ 𝐹))
5 nss 3648 . . . . . . . 8 𝑥 ⊆ {∅} ↔ ∃𝑦(𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅}))
6 simpll 789 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝐹 ∈ (Fil‘ 𝐹))
7 ssel2 3583 . . . . . . . . . . . . . . . . 17 ((𝑥 ⊆ (𝐹 ∪ {∅}) ∧ 𝑦𝑥) → 𝑦 ∈ (𝐹 ∪ {∅}))
87adantll 749 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦𝑥) → 𝑦 ∈ (𝐹 ∪ {∅}))
9 elun 3737 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝐹 ∪ {∅}) ↔ (𝑦𝐹𝑦 ∈ {∅}))
108, 9sylib 208 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦𝑥) → (𝑦𝐹𝑦 ∈ {∅}))
1110orcomd 403 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦𝑥) → (𝑦 ∈ {∅} ∨ 𝑦𝐹))
1211ord 392 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦𝑥) → (¬ 𝑦 ∈ {∅} → 𝑦𝐹))
1312impr 648 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑦𝐹)
14 uniss 4431 . . . . . . . . . . . . . 14 (𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 (𝐹 ∪ {∅}))
1514ad2antlr 762 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑥 (𝐹 ∪ {∅}))
16 uniun 4429 . . . . . . . . . . . . . 14 (𝐹 ∪ {∅}) = ( 𝐹 {∅})
17 0ex 4760 . . . . . . . . . . . . . . . 16 ∅ ∈ V
1817unisn 4424 . . . . . . . . . . . . . . 15 {∅} = ∅
1918uneq2i 3748 . . . . . . . . . . . . . 14 ( 𝐹 {∅}) = ( 𝐹 ∪ ∅)
20 un0 3945 . . . . . . . . . . . . . 14 ( 𝐹 ∪ ∅) = 𝐹
2116, 19, 203eqtrri 2648 . . . . . . . . . . . . 13 𝐹 = (𝐹 ∪ {∅})
2215, 21syl6sseqr 3637 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑥 𝐹)
23 elssuni 4440 . . . . . . . . . . . . 13 (𝑦𝑥𝑦 𝑥)
2423ad2antrl 763 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑦 𝑥)
25 filss 21597 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘ 𝐹) ∧ (𝑦𝐹 𝑥 𝐹𝑦 𝑥)) → 𝑥𝐹)
266, 13, 22, 24, 25syl13anc 1325 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑥𝐹)
27 elun1 3764 . . . . . . . . . . 11 ( 𝑥𝐹 𝑥 ∈ (𝐹 ∪ {∅}))
2826, 27syl 17 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑥 ∈ (𝐹 ∪ {∅}))
2928ex 450 . . . . . . . . 9 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → ((𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})))
3029exlimdv 1858 . . . . . . . 8 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → (∃𝑦(𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})))
315, 30syl5bi 232 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → (¬ 𝑥 ⊆ {∅} → 𝑥 ∈ (𝐹 ∪ {∅})))
32 uni0b 4436 . . . . . . . 8 ( 𝑥 = ∅ ↔ 𝑥 ⊆ {∅})
33 ssun2 3761 . . . . . . . . . 10 {∅} ⊆ (𝐹 ∪ {∅})
3417snid 4186 . . . . . . . . . 10 ∅ ∈ {∅}
3533, 34sselii 3585 . . . . . . . . 9 ∅ ∈ (𝐹 ∪ {∅})
36 eleq1 2686 . . . . . . . . 9 ( 𝑥 = ∅ → ( 𝑥 ∈ (𝐹 ∪ {∅}) ↔ ∅ ∈ (𝐹 ∪ {∅})))
3735, 36mpbiri 248 . . . . . . . 8 ( 𝑥 = ∅ → 𝑥 ∈ (𝐹 ∪ {∅}))
3832, 37sylbir 225 . . . . . . 7 (𝑥 ⊆ {∅} → 𝑥 ∈ (𝐹 ∪ {∅}))
3931, 38pm2.61d2 172 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → 𝑥 ∈ (𝐹 ∪ {∅}))
4039ex 450 . . . . 5 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})))
4140alrimiv 1852 . . . 4 (𝐹 ∈ (Fil‘ 𝐹) → ∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})))
42 filin 21598 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
43 elun1 3764 . . . . . . . . . 10 ((𝑥𝑦) ∈ 𝐹 → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
4442, 43syl 17 . . . . . . . . 9 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
45443expa 1262 . . . . . . . 8 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) ∧ 𝑦𝐹) → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
4645ralrimiva 2962 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) → ∀𝑦𝐹 (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
47 elsni 4172 . . . . . . . . 9 (𝑦 ∈ {∅} → 𝑦 = ∅)
48 ineq2 3792 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑥𝑦) = (𝑥 ∩ ∅))
49 in0 3946 . . . . . . . . . . 11 (𝑥 ∩ ∅) = ∅
5048, 49syl6eq 2671 . . . . . . . . . 10 (𝑦 = ∅ → (𝑥𝑦) = ∅)
5150, 35syl6eqel 2706 . . . . . . . . 9 (𝑦 = ∅ → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
5247, 51syl 17 . . . . . . . 8 (𝑦 ∈ {∅} → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
5352rgen 2918 . . . . . . 7 𝑦 ∈ {∅} (𝑥𝑦) ∈ (𝐹 ∪ {∅})
54 ralun 3779 . . . . . . 7 ((∀𝑦𝐹 (𝑥𝑦) ∈ (𝐹 ∪ {∅}) ∧ ∀𝑦 ∈ {∅} (𝑥𝑦) ∈ (𝐹 ∪ {∅})) → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
5546, 53, 54sylancl 693 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
5655ralrimiva 2962 . . . . 5 (𝐹 ∈ (Fil‘ 𝐹) → ∀𝑥𝐹𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
57 elsni 4172 . . . . . . 7 (𝑥 ∈ {∅} → 𝑥 = ∅)
58 ineq1 3791 . . . . . . . . . 10 (𝑥 = ∅ → (𝑥𝑦) = (∅ ∩ 𝑦))
59 0in 3947 . . . . . . . . . 10 (∅ ∩ 𝑦) = ∅
6058, 59syl6eq 2671 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝑦) = ∅)
6160, 35syl6eqel 2706 . . . . . . . 8 (𝑥 = ∅ → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
6261ralrimivw 2963 . . . . . . 7 (𝑥 = ∅ → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
6357, 62syl 17 . . . . . 6 (𝑥 ∈ {∅} → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
6463rgen 2918 . . . . 5 𝑥 ∈ {∅}∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅})
65 ralun 3779 . . . . 5 ((∀𝑥𝐹𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}) ∧ ∀𝑥 ∈ {∅}∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅})) → ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
6656, 64, 65sylancl 693 . . . 4 (𝐹 ∈ (Fil‘ 𝐹) → ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
67 p0ex 4823 . . . . . 6 {∅} ∈ V
68 unexg 6924 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐹) ∧ {∅} ∈ V) → (𝐹 ∪ {∅}) ∈ V)
6967, 68mpan2 706 . . . . 5 (𝐹 ∈ (Fil‘ 𝐹) → (𝐹 ∪ {∅}) ∈ V)
70 istopg 20640 . . . . 5 ((𝐹 ∪ {∅}) ∈ V → ((𝐹 ∪ {∅}) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})) ∧ ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))))
7169, 70syl 17 . . . 4 (𝐹 ∈ (Fil‘ 𝐹) → ((𝐹 ∪ {∅}) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})) ∧ ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))))
7241, 66, 71mpbir2and 956 . . 3 (𝐹 ∈ (Fil‘ 𝐹) → (𝐹 ∪ {∅}) ∈ Top)
7321cldopn 20775 . . . . . . . 8 (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → ( 𝐹𝑥) ∈ (𝐹 ∪ {∅}))
74 elun 3737 . . . . . . . 8 (( 𝐹𝑥) ∈ (𝐹 ∪ {∅}) ↔ (( 𝐹𝑥) ∈ 𝐹 ∨ ( 𝐹𝑥) ∈ {∅}))
7573, 74sylib 208 . . . . . . 7 (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → (( 𝐹𝑥) ∈ 𝐹 ∨ ( 𝐹𝑥) ∈ {∅}))
76 elun 3737 . . . . . . . . . 10 (𝑥 ∈ (𝐹 ∪ {∅}) ↔ (𝑥𝐹𝑥 ∈ {∅}))
77 filfbas 21592 . . . . . . . . . . . . . 14 (𝐹 ∈ (Fil‘ 𝐹) → 𝐹 ∈ (fBas‘ 𝐹))
78 fbncp 21583 . . . . . . . . . . . . . 14 ((𝐹 ∈ (fBas‘ 𝐹) ∧ 𝑥𝐹) → ¬ ( 𝐹𝑥) ∈ 𝐹)
7977, 78sylan 488 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) → ¬ ( 𝐹𝑥) ∈ 𝐹)
8079pm2.21d 118 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅))
8180ex 450 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥𝐹 → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅)))
8257a1i13 27 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥 ∈ {∅} → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅)))
8381, 82jaod 395 . . . . . . . . . 10 (𝐹 ∈ (Fil‘ 𝐹) → ((𝑥𝐹𝑥 ∈ {∅}) → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅)))
8476, 83syl5bi 232 . . . . . . . . 9 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥 ∈ (𝐹 ∪ {∅}) → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅)))
8584imp 445 . . . . . . . 8 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅))
86 elsni 4172 . . . . . . . . 9 (( 𝐹𝑥) ∈ {∅} → ( 𝐹𝑥) = ∅)
87 elssuni 4440 . . . . . . . . . . . 12 (𝑥 ∈ (𝐹 ∪ {∅}) → 𝑥 (𝐹 ∪ {∅}))
8887, 21syl6sseqr 3637 . . . . . . . . . . 11 (𝑥 ∈ (𝐹 ∪ {∅}) → 𝑥 𝐹)
8988adantl 482 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → 𝑥 𝐹)
90 ssdif0 3922 . . . . . . . . . . 11 ( 𝐹𝑥 ↔ ( 𝐹𝑥) = ∅)
9190biimpri 218 . . . . . . . . . 10 (( 𝐹𝑥) = ∅ → 𝐹𝑥)
92 eqss 3603 . . . . . . . . . . 11 (𝑥 = 𝐹 ↔ (𝑥 𝐹 𝐹𝑥))
9392simplbi2 654 . . . . . . . . . 10 (𝑥 𝐹 → ( 𝐹𝑥𝑥 = 𝐹))
9489, 91, 93syl2im 40 . . . . . . . . 9 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (( 𝐹𝑥) = ∅ → 𝑥 = 𝐹))
9586, 94syl5 34 . . . . . . . 8 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (( 𝐹𝑥) ∈ {∅} → 𝑥 = 𝐹))
9685, 95orim12d 882 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → ((( 𝐹𝑥) ∈ 𝐹 ∨ ( 𝐹𝑥) ∈ {∅}) → (𝑥 = ∅ ∨ 𝑥 = 𝐹)))
9775, 96syl5 34 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → (𝑥 = ∅ ∨ 𝑥 = 𝐹)))
9897expimpd 628 . . . . 5 (𝐹 ∈ (Fil‘ 𝐹) → ((𝑥 ∈ (𝐹 ∪ {∅}) ∧ 𝑥 ∈ (Clsd‘(𝐹 ∪ {∅}))) → (𝑥 = ∅ ∨ 𝑥 = 𝐹)))
99 elin 3780 . . . . 5 (𝑥 ∈ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ↔ (𝑥 ∈ (𝐹 ∪ {∅}) ∧ 𝑥 ∈ (Clsd‘(𝐹 ∪ {∅}))))
100 vex 3193 . . . . . 6 𝑥 ∈ V
101100elpr 4176 . . . . 5 (𝑥 ∈ {∅, 𝐹} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐹))
10298, 99, 1013imtr4g 285 . . . 4 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥 ∈ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) → 𝑥 ∈ {∅, 𝐹}))
103102ssrdv 3594 . . 3 (𝐹 ∈ (Fil‘ 𝐹) → ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ⊆ {∅, 𝐹})
10421isconn2 21157 . . 3 ((𝐹 ∪ {∅}) ∈ Conn ↔ ((𝐹 ∪ {∅}) ∈ Top ∧ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ⊆ {∅, 𝐹}))
10572, 103, 104sylanbrc 697 . 2 (𝐹 ∈ (Fil‘ 𝐹) → (𝐹 ∪ {∅}) ∈ Conn)
1064, 105syl 17 1 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wral 2908  Vcvv 3190  cdif 3557  cun 3558  cin 3559  wss 3560  c0 3897  {csn 4155  {cpr 4157   cuni 4409  cfv 5857  fBascfbas 19674  Topctop 20638  Clsdccld 20760  Conncconn 21154  Filcfil 21589
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-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865  df-fbas 19683  df-top 20639  df-cld 20763  df-conn 21155  df-fil 21590
This theorem is referenced by:  ufildr  21675
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