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Theorem kqreglem2 23750
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 22919 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21adantr 480 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Top)
3 simplr 769 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (KQ‘𝐽) ∈ Reg)
4 simpll 767 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝐽 ∈ (TopOn‘𝑋))
5 simprl 771 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑧𝐽)
6 kqval.2 . . . . . . 7 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
76kqopn 23742 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹𝑧) ∈ (KQ‘𝐽))
84, 5, 7syl2anc 584 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (𝐹𝑧) ∈ (KQ‘𝐽))
9 simprr 773 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑤𝑧)
10 toponss 22933 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
114, 5, 10syl2anc 584 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑧𝑋)
1211, 9sseldd 3984 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑤𝑋)
136kqfvima 23738 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽𝑤𝑋) → (𝑤𝑧 ↔ (𝐹𝑤) ∈ (𝐹𝑧)))
144, 5, 12, 13syl3anc 1373 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (𝑤𝑧 ↔ (𝐹𝑤) ∈ (𝐹𝑧)))
159, 14mpbid 232 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (𝐹𝑤) ∈ (𝐹𝑧))
16 regsep 23342 . . . . 5 (((KQ‘𝐽) ∈ Reg ∧ (𝐹𝑧) ∈ (KQ‘𝐽) ∧ (𝐹𝑤) ∈ (𝐹𝑧)) → ∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))
173, 8, 15, 16syl3anc 1373 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → ∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))
184adantr 480 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝐽 ∈ (TopOn‘𝑋))
196kqid 23736 . . . . . . 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 771 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑛 ∈ (KQ‘𝐽))
22 cnima 23273 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑛 ∈ (KQ‘𝐽)) → (𝐹𝑛) ∈ 𝐽)
2320, 21, 22syl2anc 584 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹𝑛) ∈ 𝐽)
2412adantr 480 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑤𝑋)
25 simprrl 781 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹𝑤) ∈ 𝑛)
266kqffn 23733 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27 elpreima 7078 . . . . . . 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 713 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑤 ∈ (𝐹𝑛))
306kqtopon 23735 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
31 topontop 22919 . . . . . . . . . 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 4937 . . . . . . . . . 10 (𝑛 ∈ (KQ‘𝐽) → 𝑛 (KQ‘𝐽))
3433ad2antrl 728 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑛 (KQ‘𝐽))
35 eqid 2737 . . . . . . . . . 10 (KQ‘𝐽) = (KQ‘𝐽)
3635clscld 23055 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑛 (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽)))
3732, 34, 36syl2anc 584 . . . . . . . 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 23276 . . . . . . . 8 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ∈ (Clsd‘𝐽))
3920, 37, 38syl2anc 584 . . . . . . 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 23064 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑛 (KQ‘𝐽)) → 𝑛 ⊆ ((cls‘(KQ‘𝐽))‘𝑛))
4132, 34, 40syl2anc 584 . . . . . . . 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 6120 . . . . . . . 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 2737 . . . . . . . 8 𝐽 = 𝐽
4544clsss2 23080 . . . . . . 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 584 . . . . . 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 782 . . . . . . . 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 6120 . . . . . . . 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 480 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑧𝐽)
516kqsat 23739 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹 “ (𝐹𝑧)) = 𝑧)
5218, 50, 51syl2anc 584 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹 “ (𝐹𝑧)) = 𝑧)
5349, 52sseqtrd 4020 . . . . . 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 3994 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧)
55 eleq2 2830 . . . . . . 7 (𝑚 = (𝐹𝑛) → (𝑤𝑚𝑤 ∈ (𝐹𝑛)))
56 fveq2 6906 . . . . . . . 8 (𝑚 = (𝐹𝑛) → ((cls‘𝐽)‘𝑚) = ((cls‘𝐽)‘(𝐹𝑛)))
5756sseq1d 4015 . . . . . . 7 (𝑚 = (𝐹𝑛) → (((cls‘𝐽)‘𝑚) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧))
5855, 57anbi12d 632 . . . . . 6 (𝑚 = (𝐹𝑛) → ((𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧) ↔ (𝑤 ∈ (𝐹𝑛) ∧ ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧)))
5958rspcev 3622 . . . . 5 (((𝐹𝑛) ∈ 𝐽 ∧ (𝑤 ∈ (𝐹𝑛) ∧ ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧)) → ∃𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
6023, 29, 54, 59syl12anc 837 . . . 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 3161 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → ∃𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
6261ralrimivva 3202 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → ∀𝑧𝐽𝑤𝑧𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
63 isreg 23340 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑧𝐽𝑤𝑧𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧)))
642, 62, 63sylanbrc 583 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  wss 3951   cuni 4907  cmpt 5225  ccnv 5684  ran crn 5686  cima 5688   Fn wfn 6556  cfv 6561  (class class class)co 7431  Topctop 22899  TopOnctopon 22916  Clsdccld 23024  clsccl 23026   Cn ccn 23232  Regcreg 23317  KQckq 23701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-qtop 17552  df-top 22900  df-topon 22917  df-cld 23027  df-cls 23029  df-cn 23235  df-reg 23324  df-kq 23702
This theorem is referenced by:  kqreg  23759
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