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Theorem kqreglem2 21464
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 20646 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21adantr 481 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Top)
3 simplr 791 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (KQ‘𝐽) ∈ Reg)
4 simpll 789 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝐽 ∈ (TopOn‘𝑋))
5 simprl 793 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑧𝐽)
6 kqval.2 . . . . . . 7 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
76kqopn 21456 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹𝑧) ∈ (KQ‘𝐽))
84, 5, 7syl2anc 692 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → (𝐹𝑧) ∈ (KQ‘𝐽))
9 simprr 795 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑤𝑧)
10 toponss 20649 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
114, 5, 10syl2anc 692 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑧𝑋)
1211, 9sseldd 3588 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → 𝑤𝑋)
136kqfvima 21452 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽𝑤𝑋) → (𝑤𝑧 ↔ (𝐹𝑤) ∈ (𝐹𝑧)))
144, 5, 12, 13syl3anc 1323 . . . . . 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 21057 . . . . 5 (((KQ‘𝐽) ∈ Reg ∧ (𝐹𝑧) ∈ (KQ‘𝐽) ∧ (𝐹𝑤) ∈ (𝐹𝑧)) → ∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))
173, 8, 15, 16syl3anc 1323 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → ∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))
184adantr 481 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝐽 ∈ (TopOn‘𝑋))
196kqid 21450 . . . . . . 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 793 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑛 ∈ (KQ‘𝐽))
22 cnima 20988 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑛 ∈ (KQ‘𝐽)) → (𝐹𝑛) ∈ 𝐽)
2320, 21, 22syl2anc 692 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹𝑛) ∈ 𝐽)
2412adantr 481 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑤𝑋)
25 simprrl 803 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹𝑤) ∈ 𝑛)
266kqffn 21447 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27 elpreima 6298 . . . . . . 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 956 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑤 ∈ (𝐹𝑛))
306kqtopon 21449 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
31 topontop 20646 . . . . . . . . . 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 4438 . . . . . . . . . 10 (𝑛 ∈ (KQ‘𝐽) → 𝑛 (KQ‘𝐽))
3433ad2antrl 763 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑛 (KQ‘𝐽))
35 eqid 2621 . . . . . . . . . 10 (KQ‘𝐽) = (KQ‘𝐽)
3635clscld 20770 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑛 (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽)))
3732, 34, 36syl2anc 692 . . . . . . . 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 20991 . . . . . . . 8 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ∈ (Clsd‘𝐽))
3920, 37, 38syl2anc 692 . . . . . . 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 20779 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑛 (KQ‘𝐽)) → 𝑛 ⊆ ((cls‘(KQ‘𝐽))‘𝑛))
4132, 34, 40syl2anc 692 . . . . . . . 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 5465 . . . . . . . 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 2621 . . . . . . . 8 𝐽 = 𝐽
4544clsss2 20795 . . . . . . 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 692 . . . . . 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 804 . . . . . . . 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 5465 . . . . . . . 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 481 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → 𝑧𝐽)
516kqsat 21453 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹 “ (𝐹𝑧)) = 𝑧)
5218, 50, 51syl2anc 692 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → (𝐹 “ (𝐹𝑧)) = 𝑧)
5349, 52sseqtrd 3625 . . . . . 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 3597 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹𝑧)))) → ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧)
55 eleq2 2687 . . . . . . 7 (𝑚 = (𝐹𝑛) → (𝑤𝑚𝑤 ∈ (𝐹𝑛)))
56 fveq2 6153 . . . . . . . 8 (𝑚 = (𝐹𝑛) → ((cls‘𝐽)‘𝑚) = ((cls‘𝐽)‘(𝐹𝑛)))
5756sseq1d 3616 . . . . . . 7 (𝑚 = (𝐹𝑛) → (((cls‘𝐽)‘𝑚) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧))
5855, 57anbi12d 746 . . . . . 6 (𝑚 = (𝐹𝑛) → ((𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧) ↔ (𝑤 ∈ (𝐹𝑛) ∧ ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧)))
5958rspcev 3298 . . . . 5 (((𝐹𝑛) ∈ 𝐽 ∧ (𝑤 ∈ (𝐹𝑛) ∧ ((cls‘𝐽)‘(𝐹𝑛)) ⊆ 𝑧)) → ∃𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
6023, 29, 54, 59syl12anc 1321 . . . 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 3029 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧𝐽𝑤𝑧)) → ∃𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
6261ralrimivva 2966 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → ∀𝑧𝐽𝑤𝑧𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
63 isreg 21055 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑧𝐽𝑤𝑧𝑚𝐽 (𝑤𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧)))
642, 62, 63sylanbrc 697 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  wss 3559   cuni 4407  cmpt 4678  ccnv 5078  ran crn 5080  cima 5082   Fn wfn 5847  cfv 5852  (class class class)co 6610  Topctop 20626  TopOnctopon 20643  Clsdccld 20739  clsccl 20741   Cn ccn 20947  Regcreg 21032  KQckq 21415
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-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-qtop 16095  df-top 20627  df-topon 20644  df-cld 20742  df-cls 20744  df-cn 20950  df-reg 21039  df-kq 21416
This theorem is referenced by:  kqreg  21473
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