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Theorem kqreglem2 21766
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 20938 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21adantr 466 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Top)
3 simplr 752 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (KQ‘𝐽) ∈ Reg)
4 simpll 750 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝐽 ∈ (TopOn‘𝑋))
5 simprl 754 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑧𝐽)
6 kqval.2 . . . . . . 7 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
76kqopn 21758 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹𝑧) ∈ (KQ‘𝐽))
84, 5, 7syl2anc 573 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (𝐹𝑧) ∈ (KQ‘𝐽))
9 simprr 756 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑤𝑧)
10 toponss 20952 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
114, 5, 10syl2anc 573 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑧𝑋)
1211, 9sseldd 3753 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑤𝑋)
136kqfvima 21754 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽𝑤𝑋) → (𝑤𝑧 ↔ (𝐹𝑤) ∈ (𝐹𝑧)))
144, 5, 12, 13syl3anc 1476 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (𝑤𝑧 ↔ (𝐹𝑤) ∈ (𝐹𝑧)))
159, 14mpbid 222 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (𝐹𝑤) ∈ (𝐹𝑧))
16 regsep 21359 . . . . 5 (((KQ‘𝐽) ∈ Reg ∧ (𝐹𝑧) ∈ (KQ‘𝐽) ∧ (𝐹𝑤) ∈ (𝐹𝑧)) → ∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))
173, 8, 15, 16syl3anc 1476 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → ∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))
184adantr 466 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝐽 ∈ (TopOn‘𝑋))
196kqid 21752 . . . . . . 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 754 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑛 ∈ (KQ‘𝐽))
22 cnima 21290 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑛 ∈ (KQ‘𝐽)) → (𝐹𝑛) ∈ 𝐽)
2320, 21, 22syl2anc 573 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹𝑛) ∈ 𝐽)
2412adantr 466 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑤𝑋)
25 simprrl 766 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹𝑤) ∈ 𝑛)
266kqffn 21749 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27 elpreima 6480 . . . . . . 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 692 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑤 ∈ (𝐹𝑛))
306kqtopon 21751 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
31 topontop 20938 . . . . . . . . . 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 4603 . . . . . . . . . 10 (𝑛 ∈ (KQ‘𝐽) → 𝑛 (KQ‘𝐽))
3433ad2antrl 707 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑛 (KQ‘𝐽))
35 eqid 2771 . . . . . . . . . 10 (KQ‘𝐽) = (KQ‘𝐽)
3635clscld 21072 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑛 (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽)))
3732, 34, 36syl2anc 573 . . . . . . . 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 21293 . . . . . . . 8 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ∈ (Clsd‘𝐽))
3920, 37, 38syl2anc 573 . . . . . . 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 21081 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑛 (KQ‘𝐽)) → 𝑛 ⊆ ((cls‘(KQ‘𝐽))‘𝑛))
4132, 34, 40syl2anc 573 . . . . . . . 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 5642 . . . . . . . 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 2771 . . . . . . . 8 𝐽 = 𝐽
4544clsss2 21097 . . . . . . 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 573 . . . . . 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 767 . . . . . . . 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 5642 . . . . . . . 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 466 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑧𝐽)
516kqsat 21755 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹 “ (𝐹𝑧)) = 𝑧)
5218, 50, 51syl2anc 573 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹 “ (𝐹𝑧)) = 𝑧)
5349, 52sseqtrd 3790 . . . . . 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 3762 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧)
55 eleq2 2839 . . . . . . 7 (𝑚 = (𝐹𝑛) → (𝑤𝑚𝑤 ∈ (𝐹𝑛)))
56 fveq2 6332 . . . . . . . 8 (𝑚 = (𝐹𝑛) → ((cls‘𝐽)‘𝑚) = ((cls‘𝐽)‘(𝐹𝑛)))
5756sseq1d 3781 . . . . . . 7 (𝑚 = (𝐹𝑛) → (((cls‘𝐽)‘𝑚) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧))
5855, 57anbi12d 616 . . . . . 6 (𝑚 = (𝐹𝑛) → ((𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧) ↔ (𝑤 ∈ (𝐹𝑛) ∧ ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧)))
5958rspcev 3460 . . . . 5 (((𝐹𝑛) ∈ 𝐽 ∧ (𝑤 ∈ (𝐹𝑛) ∧ ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧)) → ∃𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
6023, 29, 54, 59syl12anc 1474 . . . 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 3183 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → ∃𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
6261ralrimivva 3120 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → ∀𝑧𝐽𝑤𝑧𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
63 isreg 21357 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑧𝐽𝑤𝑧𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧)))
642, 62, 63sylanbrc 572 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  {crab 3065  wss 3723   cuni 4574  cmpt 4863  ccnv 5248  ran crn 5250  cima 5252   Fn wfn 6026  cfv 6031  (class class class)co 6793  Topctop 20918  TopOnctopon 20935  Clsdccld 21041  clsccl 21043   Cn ccn 21249  Regcreg 21334  KQckq 21717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-qtop 16375  df-top 20919  df-topon 20936  df-cld 21044  df-cls 21046  df-cn 21252  df-reg 21341  df-kq 21718
This theorem is referenced by:  kqreg  21775
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