MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqreglem2 Structured version   Visualization version   GIF version

Theorem kqreglem2 21965
Description: If the Kolmogorov quotient of a space is regular then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqreglem2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqreglem2
Dummy variables 𝑚 𝑛 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 21136 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21adantr 474 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Top)
3 simplr 759 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (KQ‘𝐽) ∈ Reg)
4 simpll 757 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝐽 ∈ (TopOn‘𝑋))
5 simprl 761 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑧𝐽)
6 kqval.2 . . . . . . 7 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
76kqopn 21957 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹𝑧) ∈ (KQ‘𝐽))
84, 5, 7syl2anc 579 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (𝐹𝑧) ∈ (KQ‘𝐽))
9 simprr 763 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑤𝑧)
10 toponss 21150 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
114, 5, 10syl2anc 579 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑧𝑋)
1211, 9sseldd 3822 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑤𝑋)
136kqfvima 21953 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽𝑤𝑋) → (𝑤𝑧 ↔ (𝐹𝑤) ∈ (𝐹𝑧)))
144, 5, 12, 13syl3anc 1439 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (𝑤𝑧 ↔ (𝐹𝑤) ∈ (𝐹𝑧)))
159, 14mpbid 224 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (𝐹𝑤) ∈ (𝐹𝑧))
16 regsep 21557 . . . . 5 (((KQ‘𝐽) ∈ Reg ∧ (𝐹𝑧) ∈ (KQ‘𝐽) ∧ (𝐹𝑤) ∈ (𝐹𝑧)) → ∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))
173, 8, 15, 16syl3anc 1439 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → ∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))
184adantr 474 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝐽 ∈ (TopOn‘𝑋))
196kqid 21951 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
2018, 19syl 17 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
21 simprl 761 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑛 ∈ (KQ‘𝐽))
22 cnima 21488 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑛 ∈ (KQ‘𝐽)) → (𝐹𝑛) ∈ 𝐽)
2320, 21, 22syl2anc 579 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹𝑛) ∈ 𝐽)
2412adantr 474 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑤𝑋)
25 simprrl 771 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹𝑤) ∈ 𝑛)
266kqffn 21948 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27 elpreima 6602 . . . . . . 7 (𝐹 Fn 𝑋 → (𝑤 ∈ (𝐹𝑛) ↔ (𝑤𝑋 ∧ (𝐹𝑤) ∈ 𝑛)))
2818, 26, 273syl 18 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝑤 ∈ (𝐹𝑛) ↔ (𝑤𝑋 ∧ (𝐹𝑤) ∈ 𝑛)))
2924, 25, 28mpbir2and 703 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑤 ∈ (𝐹𝑛))
306kqtopon 21950 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
31 topontop 21136 . . . . . . . . . 10 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
3218, 30, 313syl 18 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (KQ‘𝐽) ∈ Top)
33 elssuni 4704 . . . . . . . . . 10 (𝑛 ∈ (KQ‘𝐽) → 𝑛 (KQ‘𝐽))
3433ad2antrl 718 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑛 (KQ‘𝐽))
35 eqid 2778 . . . . . . . . . 10 (KQ‘𝐽) = (KQ‘𝐽)
3635clscld 21270 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑛 (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽)))
3732, 34, 36syl2anc 579 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽)))
38 cnclima 21491 . . . . . . . 8 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ∈ (Clsd‘𝐽))
3920, 37, 38syl2anc 579 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ∈ (Clsd‘𝐽))
4035sscls 21279 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑛 (KQ‘𝐽)) → 𝑛 ⊆ ((cls‘(KQ‘𝐽))‘𝑛))
4132, 34, 40syl2anc 579 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑛 ⊆ ((cls‘(KQ‘𝐽))‘𝑛))
42 imass2 5757 . . . . . . . 8 (𝑛 ⊆ ((cls‘(KQ‘𝐽))‘𝑛) → (𝐹𝑛) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)))
4341, 42syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹𝑛) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)))
44 eqid 2778 . . . . . . . 8 𝐽 = 𝐽
4544clsss2 21295 . . . . . . 7 (((𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ∈ (Clsd‘𝐽) ∧ (𝐹𝑛) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛))) → ((cls‘𝐽)‘(𝐹𝑛)) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)))
4639, 43, 45syl2anc 579 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → ((cls‘𝐽)‘(𝐹𝑛)) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)))
47 simprrr 772 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧))
48 imass2 5757 . . . . . . . 8 (((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ⊆ (𝐹 “ (𝐹𝑧)))
4947, 48syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ⊆ (𝐹 “ (𝐹𝑧)))
505adantr 474 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑧𝐽)
516kqsat 21954 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹 “ (𝐹𝑧)) = 𝑧)
5218, 50, 51syl2anc 579 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹 “ (𝐹𝑧)) = 𝑧)
5349, 52sseqtrd 3860 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ⊆ 𝑧)
5446, 53sstrd 3831 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧)
55 eleq2 2848 . . . . . . 7 (𝑚 = (𝐹𝑛) → (𝑤𝑚𝑤 ∈ (𝐹𝑛)))
56 fveq2 6448 . . . . . . . 8 (𝑚 = (𝐹𝑛) → ((cls‘𝐽)‘𝑚) = ((cls‘𝐽)‘(𝐹𝑛)))
5756sseq1d 3851 . . . . . . 7 (𝑚 = (𝐹𝑛) → (((cls‘𝐽)‘𝑚) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧))
5855, 57anbi12d 624 . . . . . 6 (𝑚 = (𝐹𝑛) → ((𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧) ↔ (𝑤 ∈ (𝐹𝑛) ∧ ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧)))
5958rspcev 3511 . . . . 5 (((𝐹𝑛) ∈ 𝐽 ∧ (𝑤 ∈ (𝐹𝑛) ∧ ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧)) → ∃𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
6023, 29, 54, 59syl12anc 827 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → ∃𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
6117, 60rexlimddv 3218 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → ∃𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
6261ralrimivva 3153 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → ∀𝑧𝐽𝑤𝑧𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
63 isreg 21555 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑧𝐽𝑤𝑧𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧)))
642, 62, 63sylanbrc 578 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wral 3090  wrex 3091  {crab 3094  wss 3792   cuni 4673  cmpt 4967  ccnv 5356  ran crn 5358  cima 5360   Fn wfn 6132  cfv 6137  (class class class)co 6924  Topctop 21116  TopOnctopon 21133  Clsdccld 21239  clsccl 21241   Cn ccn 21447  Regcreg 21532  KQckq 21916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-map 8144  df-qtop 16564  df-top 21117  df-topon 21134  df-cld 21242  df-cls 21244  df-cn 21450  df-reg 21539  df-kq 21917
This theorem is referenced by:  kqreg  21974
  Copyright terms: Public domain W3C validator