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Theorem reghmph 22405
 Description: Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
reghmph (𝐽𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))

Proof of Theorem reghmph
Dummy variables 𝑤 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmph 22388 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 4293 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 hmeocn 22372 . . . . . . . 8 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
43adantl 485 . . . . . . 7 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
5 cntop2 21853 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
64, 5syl 17 . . . . . 6 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Top)
7 simpll 766 . . . . . . . . 9 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝐽 ∈ Reg)
84adantr 484 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑓 ∈ (𝐽 Cn 𝐾))
9 simprl 770 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑥𝐾)
10 cnima 21877 . . . . . . . . . 10 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥𝐾) → (𝑓𝑥) ∈ 𝐽)
118, 9, 10syl2anc 587 . . . . . . . . 9 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → (𝑓𝑥) ∈ 𝐽)
12 eqid 2824 . . . . . . . . . . . . 13 𝐽 = 𝐽
13 eqid 2824 . . . . . . . . . . . . 13 𝐾 = 𝐾
1412, 13hmeof1o 22376 . . . . . . . . . . . 12 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓: 𝐽1-1-onto 𝐾)
1514ad2antlr 726 . . . . . . . . . . 11 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑓: 𝐽1-1-onto 𝐾)
16 f1ocnv 6618 . . . . . . . . . . 11 (𝑓: 𝐽1-1-onto 𝐾𝑓: 𝐾1-1-onto 𝐽)
17 f1ofn 6607 . . . . . . . . . . 11 (𝑓: 𝐾1-1-onto 𝐽𝑓 Fn 𝐾)
1815, 16, 173syl 18 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑓 Fn 𝐾)
19 elssuni 4854 . . . . . . . . . . 11 (𝑥𝐾𝑥 𝐾)
2019ad2antrl 727 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑥 𝐾)
21 simprr 772 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑦𝑥)
22 fnfvima 6987 . . . . . . . . . 10 ((𝑓 Fn 𝐾𝑥 𝐾𝑦𝑥) → (𝑓𝑦) ∈ (𝑓𝑥))
2318, 20, 21, 22syl3anc 1368 . . . . . . . . 9 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → (𝑓𝑦) ∈ (𝑓𝑥))
24 regsep 21946 . . . . . . . . 9 ((𝐽 ∈ Reg ∧ (𝑓𝑥) ∈ 𝐽 ∧ (𝑓𝑦) ∈ (𝑓𝑥)) → ∃𝑤𝐽 ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
257, 11, 23, 24syl3anc 1368 . . . . . . . 8 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → ∃𝑤𝐽 ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
26 simpllr 775 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓 ∈ (𝐽Homeo𝐾))
27 simprl 770 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑤𝐽)
28 hmeoima 22377 . . . . . . . . . 10 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤𝐽) → (𝑓𝑤) ∈ 𝐾)
2926, 27, 28syl2anc 587 . . . . . . . . 9 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓𝑤) ∈ 𝐾)
3020, 21sseldd 3954 . . . . . . . . . . . 12 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑦 𝐾)
3130adantr 484 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 𝐾)
32 simprrl 780 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓𝑦) ∈ 𝑤)
3318adantr 484 . . . . . . . . . . . 12 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓 Fn 𝐾)
34 elpreima 6819 . . . . . . . . . . . 12 (𝑓 Fn 𝐾 → (𝑦 ∈ (𝑓𝑤) ↔ (𝑦 𝐾 ∧ (𝑓𝑦) ∈ 𝑤)))
3533, 34syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑦 ∈ (𝑓𝑤) ↔ (𝑦 𝐾 ∧ (𝑓𝑦) ∈ 𝑤)))
3631, 32, 35mpbir2and 712 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 ∈ (𝑓𝑤))
37 imacnvcnv 6050 . . . . . . . . . 10 (𝑓𝑤) = (𝑓𝑤)
3836, 37eleqtrdi 2926 . . . . . . . . 9 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 ∈ (𝑓𝑤))
39 elssuni 4854 . . . . . . . . . . . 12 (𝑤𝐽𝑤 𝐽)
4039ad2antrl 727 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑤 𝐽)
4112hmeocls 22380 . . . . . . . . . . 11 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 𝐽) → ((cls‘𝐾)‘(𝑓𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
4226, 40, 41syl2anc 587 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐾)‘(𝑓𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
43 simprrr 781 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥))
4415adantr 484 . . . . . . . . . . . . 13 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓: 𝐽1-1-onto 𝐾)
45 f1ofun 6608 . . . . . . . . . . . . 13 (𝑓: 𝐽1-1-onto 𝐾 → Fun 𝑓)
4644, 45syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → Fun 𝑓)
477adantr 484 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝐽 ∈ Reg)
48 regtop 21945 . . . . . . . . . . . . . . 15 (𝐽 ∈ Reg → 𝐽 ∈ Top)
4947, 48syl 17 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝐽 ∈ Top)
5012clsss3 21671 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑤 𝐽) → ((cls‘𝐽)‘𝑤) ⊆ 𝐽)
5149, 40, 50syl2anc 587 . . . . . . . . . . . . 13 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ 𝐽)
52 f1odm 6610 . . . . . . . . . . . . . 14 (𝑓: 𝐽1-1-onto 𝐾 → dom 𝑓 = 𝐽)
5344, 52syl 17 . . . . . . . . . . . . 13 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → dom 𝑓 = 𝐽)
5451, 53sseqtrrd 3994 . . . . . . . . . . . 12 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓)
55 funimass3 6815 . . . . . . . . . . . 12 ((Fun 𝑓 ∧ ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
5646, 54, 55syl2anc 587 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
5743, 56mpbird 260 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥)
5842, 57eqsstrd 3991 . . . . . . . . 9 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)
59 eleq2 2904 . . . . . . . . . . 11 (𝑧 = (𝑓𝑤) → (𝑦𝑧𝑦 ∈ (𝑓𝑤)))
60 fveq2 6661 . . . . . . . . . . . 12 (𝑧 = (𝑓𝑤) → ((cls‘𝐾)‘𝑧) = ((cls‘𝐾)‘(𝑓𝑤)))
6160sseq1d 3984 . . . . . . . . . . 11 (𝑧 = (𝑓𝑤) → (((cls‘𝐾)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥))
6259, 61anbi12d 633 . . . . . . . . . 10 (𝑧 = (𝑓𝑤) → ((𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ∈ (𝑓𝑤) ∧ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)))
6362rspcev 3609 . . . . . . . . 9 (((𝑓𝑤) ∈ 𝐾 ∧ (𝑦 ∈ (𝑓𝑤) ∧ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6429, 38, 58, 63syl12anc 835 . . . . . . . 8 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6525, 64rexlimddv 3283 . . . . . . 7 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6665ralrimivva 3186 . . . . . 6 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → ∀𝑥𝐾𝑦𝑥𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
67 isreg 21944 . . . . . 6 (𝐾 ∈ Reg ↔ (𝐾 ∈ Top ∧ ∀𝑥𝐾𝑦𝑥𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥)))
686, 66, 67sylanbrc 586 . . . . 5 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Reg)
6968expcom 417 . . . 4 (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
7069exlimiv 1932 . . 3 (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
712, 70sylbi 220 . 2 ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
721, 71sylbi 220 1 (𝐽𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  ∃wrex 3134   ⊆ wss 3919  ∅c0 4276  ∪ cuni 4824   class class class wbr 5052  ◡ccnv 5541  dom cdm 5542   “ cima 5545  Fun wfun 6337   Fn wfn 6338  –1-1-onto→wf1o 6342  ‘cfv 6343  (class class class)co 7149  Topctop 21505  clsccl 21630   Cn ccn 21836  Regcreg 21921  Homeochmeo 22365   ≃ chmph 22366 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-1o 8098  df-map 8404  df-top 21506  df-topon 21523  df-cld 21631  df-cls 21633  df-cn 21839  df-reg 21928  df-hmeo 22367  df-hmph 22368 This theorem is referenced by: (None)
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