Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  filcon Structured version   Visualization version   GIF version

Theorem filcon 21497
 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
filcon (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Con)

Proof of Theorem filcon
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 21495 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
32fveq2d 6107 . . 3 (𝐹 ∈ (Fil‘𝑋) → (Fil‘ 𝐹) = (Fil‘𝑋))
41, 3eleqtrrd 2691 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (Fil‘ 𝐹))
5 nss 3626 . . . . . . . 8 𝑥 ⊆ {∅} ↔ ∃𝑦(𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅}))
6 simpll 786 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝐹 ∈ (Fil‘ 𝐹))
7 ssel2 3563 . . . . . . . . . . . . . . . . 17 ((𝑥 ⊆ (𝐹 ∪ {∅}) ∧ 𝑦𝑥) → 𝑦 ∈ (𝐹 ∪ {∅}))
87adantll 746 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦𝑥) → 𝑦 ∈ (𝐹 ∪ {∅}))
9 elun 3715 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝐹 ∪ {∅}) ↔ (𝑦𝐹𝑦 ∈ {∅}))
108, 9sylib 207 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦𝑥) → (𝑦𝐹𝑦 ∈ {∅}))
1110orcomd 402 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦𝑥) → (𝑦 ∈ {∅} ∨ 𝑦𝐹))
1211ord 391 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦𝑥) → (¬ 𝑦 ∈ {∅} → 𝑦𝐹))
1312impr 647 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑦𝐹)
14 uniss 4394 . . . . . . . . . . . . . 14 (𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 (𝐹 ∪ {∅}))
1514ad2antlr 759 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑥 (𝐹 ∪ {∅}))
16 uniun 4392 . . . . . . . . . . . . . 14 (𝐹 ∪ {∅}) = ( 𝐹 {∅})
17 0ex 4718 . . . . . . . . . . . . . . . 16 ∅ ∈ V
1817unisn 4387 . . . . . . . . . . . . . . 15 {∅} = ∅
1918uneq2i 3726 . . . . . . . . . . . . . 14 ( 𝐹 {∅}) = ( 𝐹 ∪ ∅)
20 un0 3919 . . . . . . . . . . . . . 14 ( 𝐹 ∪ ∅) = 𝐹
2116, 19, 203eqtrri 2637 . . . . . . . . . . . . 13 𝐹 = (𝐹 ∪ {∅})
2215, 21syl6sseqr 3615 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑥 𝐹)
23 elssuni 4403 . . . . . . . . . . . . 13 (𝑦𝑥𝑦 𝑥)
2423ad2antrl 760 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑦 𝑥)
25 filss 21467 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘ 𝐹) ∧ (𝑦𝐹 𝑥 𝐹𝑦 𝑥)) → 𝑥𝐹)
266, 13, 22, 24, 25syl13anc 1320 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑥𝐹)
27 elun1 3742 . . . . . . . . . . 11 ( 𝑥𝐹 𝑥 ∈ (𝐹 ∪ {∅}))
2826, 27syl 17 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑥 ∈ (𝐹 ∪ {∅}))
2928ex 449 . . . . . . . . 9 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → ((𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})))
3029exlimdv 1848 . . . . . . . 8 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → (∃𝑦(𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})))
315, 30syl5bi 231 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → (¬ 𝑥 ⊆ {∅} → 𝑥 ∈ (𝐹 ∪ {∅})))
32 uni0b 4399 . . . . . . . 8 ( 𝑥 = ∅ ↔ 𝑥 ⊆ {∅})
33 ssun2 3739 . . . . . . . . . 10 {∅} ⊆ (𝐹 ∪ {∅})
3417snid 4155 . . . . . . . . . 10 ∅ ∈ {∅}
3533, 34sselii 3565 . . . . . . . . 9 ∅ ∈ (𝐹 ∪ {∅})
36 eleq1 2676 . . . . . . . . 9 ( 𝑥 = ∅ → ( 𝑥 ∈ (𝐹 ∪ {∅}) ↔ ∅ ∈ (𝐹 ∪ {∅})))
3735, 36mpbiri 247 . . . . . . . 8 ( 𝑥 = ∅ → 𝑥 ∈ (𝐹 ∪ {∅}))
3832, 37sylbir 224 . . . . . . 7 (𝑥 ⊆ {∅} → 𝑥 ∈ (𝐹 ∪ {∅}))
3931, 38pm2.61d2 171 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → 𝑥 ∈ (𝐹 ∪ {∅}))
4039ex 449 . . . . 5 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})))
4140alrimiv 1842 . . . 4 (𝐹 ∈ (Fil‘ 𝐹) → ∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})))
42 filin 21468 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
43 elun1 3742 . . . . . . . . . 10 ((𝑥𝑦) ∈ 𝐹 → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
4442, 43syl 17 . . . . . . . . 9 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
45443expa 1257 . . . . . . . 8 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) ∧ 𝑦𝐹) → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
4645ralrimiva 2949 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) → ∀𝑦𝐹 (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
47 elsni 4142 . . . . . . . . 9 (𝑦 ∈ {∅} → 𝑦 = ∅)
48 ineq2 3770 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑥𝑦) = (𝑥 ∩ ∅))
49 in0 3920 . . . . . . . . . . 11 (𝑥 ∩ ∅) = ∅
5048, 49syl6eq 2660 . . . . . . . . . 10 (𝑦 = ∅ → (𝑥𝑦) = ∅)
5150, 35syl6eqel 2696 . . . . . . . . 9 (𝑦 = ∅ → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
5247, 51syl 17 . . . . . . . 8 (𝑦 ∈ {∅} → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
5352rgen 2906 . . . . . . 7 𝑦 ∈ {∅} (𝑥𝑦) ∈ (𝐹 ∪ {∅})
54 ralun 3757 . . . . . . 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 2949 . . . . 5 (𝐹 ∈ (Fil‘ 𝐹) → ∀𝑥𝐹𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
57 elsni 4142 . . . . . . 7 (𝑥 ∈ {∅} → 𝑥 = ∅)
58 ineq1 3769 . . . . . . . . . 10 (𝑥 = ∅ → (𝑥𝑦) = (∅ ∩ 𝑦))
59 0in 3921 . . . . . . . . . 10 (∅ ∩ 𝑦) = ∅
6058, 59syl6eq 2660 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝑦) = ∅)
6160, 35syl6eqel 2696 . . . . . . . 8 (𝑥 = ∅ → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
6261ralrimivw 2950 . . . . . . 7 (𝑥 = ∅ → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
6357, 62syl 17 . . . . . 6 (𝑥 ∈ {∅} → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
6463rgen 2906 . . . . 5 𝑥 ∈ {∅}∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅})
65 ralun 3757 . . . . 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 4779 . . . . . 6 {∅} ∈ V
68 unexg 6857 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐹) ∧ {∅} ∈ V) → (𝐹 ∪ {∅}) ∈ V)
6967, 68mpan2 703 . . . . 5 (𝐹 ∈ (Fil‘ 𝐹) → (𝐹 ∪ {∅}) ∈ V)
70 istopg 20525 . . . . 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 959 . . 3 (𝐹 ∈ (Fil‘ 𝐹) → (𝐹 ∪ {∅}) ∈ Top)
7321cldopn 20645 . . . . . . . 8 (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → ( 𝐹𝑥) ∈ (𝐹 ∪ {∅}))
74 elun 3715 . . . . . . . 8 (( 𝐹𝑥) ∈ (𝐹 ∪ {∅}) ↔ (( 𝐹𝑥) ∈ 𝐹 ∨ ( 𝐹𝑥) ∈ {∅}))
7573, 74sylib 207 . . . . . . 7 (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → (( 𝐹𝑥) ∈ 𝐹 ∨ ( 𝐹𝑥) ∈ {∅}))
76 elun 3715 . . . . . . . . . 10 (𝑥 ∈ (𝐹 ∪ {∅}) ↔ (𝑥𝐹𝑥 ∈ {∅}))
77 filfbas 21462 . . . . . . . . . . . . . 14 (𝐹 ∈ (Fil‘ 𝐹) → 𝐹 ∈ (fBas‘ 𝐹))
78 fbncp 21453 . . . . . . . . . . . . . 14 ((𝐹 ∈ (fBas‘ 𝐹) ∧ 𝑥𝐹) → ¬ ( 𝐹𝑥) ∈ 𝐹)
7977, 78sylan 487 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) → ¬ ( 𝐹𝑥) ∈ 𝐹)
8079pm2.21d 117 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅))
8180ex 449 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥𝐹 → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅)))
8257a1i13 27 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥 ∈ {∅} → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅)))
8381, 82jaod 394 . . . . . . . . . 10 (𝐹 ∈ (Fil‘ 𝐹) → ((𝑥𝐹𝑥 ∈ {∅}) → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅)))
8476, 83syl5bi 231 . . . . . . . . 9 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥 ∈ (𝐹 ∪ {∅}) → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅)))
8584imp 444 . . . . . . . 8 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅))
86 elsni 4142 . . . . . . . . 9 (( 𝐹𝑥) ∈ {∅} → ( 𝐹𝑥) = ∅)
87 elssuni 4403 . . . . . . . . . . . 12 (𝑥 ∈ (𝐹 ∪ {∅}) → 𝑥 (𝐹 ∪ {∅}))
8887, 21syl6sseqr 3615 . . . . . . . . . . 11 (𝑥 ∈ (𝐹 ∪ {∅}) → 𝑥 𝐹)
8988adantl 481 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → 𝑥 𝐹)
90 ssdif0 3896 . . . . . . . . . . 11 ( 𝐹𝑥 ↔ ( 𝐹𝑥) = ∅)
9190biimpri 217 . . . . . . . . . 10 (( 𝐹𝑥) = ∅ → 𝐹𝑥)
92 eqss 3583 . . . . . . . . . . 11 (𝑥 = 𝐹 ↔ (𝑥 𝐹 𝐹𝑥))
9392simplbi2 653 . . . . . . . . . 10 (𝑥 𝐹 → ( 𝐹𝑥𝑥 = 𝐹))
9489, 91, 93syl2im 39 . . . . . . . . 9 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (( 𝐹𝑥) = ∅ → 𝑥 = 𝐹))
9586, 94syl5 33 . . . . . . . 8 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (( 𝐹𝑥) ∈ {∅} → 𝑥 = 𝐹))
9685, 95orim12d 879 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → ((( 𝐹𝑥) ∈ 𝐹 ∨ ( 𝐹𝑥) ∈ {∅}) → (𝑥 = ∅ ∨ 𝑥 = 𝐹)))
9775, 96syl5 33 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → (𝑥 = ∅ ∨ 𝑥 = 𝐹)))
9897expimpd 627 . . . . 5 (𝐹 ∈ (Fil‘ 𝐹) → ((𝑥 ∈ (𝐹 ∪ {∅}) ∧ 𝑥 ∈ (Clsd‘(𝐹 ∪ {∅}))) → (𝑥 = ∅ ∨ 𝑥 = 𝐹)))
99 elin 3758 . . . . 5 (𝑥 ∈ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ↔ (𝑥 ∈ (𝐹 ∪ {∅}) ∧ 𝑥 ∈ (Clsd‘(𝐹 ∪ {∅}))))
100 vex 3176 . . . . . 6 𝑥 ∈ V
101100elpr 4146 . . . . 5 (𝑥 ∈ {∅, 𝐹} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐹))
10298, 99, 1013imtr4g 284 . . . 4 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥 ∈ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) → 𝑥 ∈ {∅, 𝐹}))
103102ssrdv 3574 . . 3 (𝐹 ∈ (Fil‘ 𝐹) → ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ⊆ {∅, 𝐹})
10421iscon2 21027 . . 3 ((𝐹 ∪ {∅}) ∈ Con ↔ ((𝐹 ∪ {∅}) ∈ Top ∧ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ⊆ {∅, 𝐹}))
10572, 103, 104sylanbrc 695 . 2 (𝐹 ∈ (Fil‘ 𝐹) → (𝐹 ∪ {∅}) ∈ Con)
1064, 105syl 17 1 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Con)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127  ∪ cuni 4372  ‘cfv 5804  fBascfbas 19555  Topctop 20517  Clsdccld 20630  Conccon 21024  Filcfil 21459 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-fbas 19564  df-top 20521  df-cld 20633  df-con 21025  df-fil 21460 This theorem is referenced by:  ufildr  21545
 Copyright terms: Public domain W3C validator