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

Theorem reghmph 21519
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 21502 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 3912 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 hmeocn 21486 . . . . . . . 8 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
43adantl 482 . . . . . . 7 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
5 cntop2 20968 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
64, 5syl 17 . . . . . 6 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Top)
7 simpll 789 . . . . . . . . 9 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝐽 ∈ Reg)
84adantr 481 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑓 ∈ (𝐽 Cn 𝐾))
9 simprl 793 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑥𝐾)
10 cnima 20992 . . . . . . . . . 10 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥𝐾) → (𝑓𝑥) ∈ 𝐽)
118, 9, 10syl2anc 692 . . . . . . . . 9 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → (𝑓𝑥) ∈ 𝐽)
12 eqid 2621 . . . . . . . . . . . . 13 𝐽 = 𝐽
13 eqid 2621 . . . . . . . . . . . . 13 𝐾 = 𝐾
1412, 13hmeof1o 21490 . . . . . . . . . . . 12 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓: 𝐽1-1-onto 𝐾)
1514ad2antlr 762 . . . . . . . . . . 11 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑓: 𝐽1-1-onto 𝐾)
16 f1ocnv 6111 . . . . . . . . . . 11 (𝑓: 𝐽1-1-onto 𝐾𝑓: 𝐾1-1-onto 𝐽)
17 f1ofn 6100 . . . . . . . . . . 11 (𝑓: 𝐾1-1-onto 𝐽𝑓 Fn 𝐾)
1815, 16, 173syl 18 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑓 Fn 𝐾)
19 elssuni 4438 . . . . . . . . . . 11 (𝑥𝐾𝑥 𝐾)
2019ad2antrl 763 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑥 𝐾)
21 simprr 795 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑦𝑥)
22 fnfvima 6456 . . . . . . . . . 10 ((𝑓 Fn 𝐾𝑥 𝐾𝑦𝑥) → (𝑓𝑦) ∈ (𝑓𝑥))
2318, 20, 21, 22syl3anc 1323 . . . . . . . . 9 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → (𝑓𝑦) ∈ (𝑓𝑥))
24 regsep 21061 . . . . . . . . 9 ((𝐽 ∈ Reg ∧ (𝑓𝑥) ∈ 𝐽 ∧ (𝑓𝑦) ∈ (𝑓𝑥)) → ∃𝑤𝐽 ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
257, 11, 23, 24syl3anc 1323 . . . . . . . 8 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → ∃𝑤𝐽 ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
26 simpllr 798 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓 ∈ (𝐽Homeo𝐾))
27 simprl 793 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑤𝐽)
28 hmeoima 21491 . . . . . . . . . 10 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤𝐽) → (𝑓𝑤) ∈ 𝐾)
2926, 27, 28syl2anc 692 . . . . . . . . 9 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓𝑤) ∈ 𝐾)
3020, 21sseldd 3588 . . . . . . . . . . . 12 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑦 𝐾)
3130adantr 481 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 𝐾)
32 simprrl 803 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓𝑦) ∈ 𝑤)
3318adantr 481 . . . . . . . . . . . 12 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓 Fn 𝐾)
34 elpreima 6298 . . . . . . . . . . . 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 956 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 ∈ (𝑓𝑤))
37 imacnvcnv 5563 . . . . . . . . . 10 (𝑓𝑤) = (𝑓𝑤)
3836, 37syl6eleq 2708 . . . . . . . . 9 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 ∈ (𝑓𝑤))
39 elssuni 4438 . . . . . . . . . . . 12 (𝑤𝐽𝑤 𝐽)
4039ad2antrl 763 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑤 𝐽)
4112hmeocls 21494 . . . . . . . . . . 11 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 𝐽) → ((cls‘𝐾)‘(𝑓𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
4226, 40, 41syl2anc 692 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐾)‘(𝑓𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
43 simprrr 804 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥))
4415adantr 481 . . . . . . . . . . . . 13 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓: 𝐽1-1-onto 𝐾)
45 f1ofun 6101 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝐽 ∈ Reg)
48 regtop 21060 . . . . . . . . . . . . . . 15 (𝐽 ∈ Reg → 𝐽 ∈ Top)
4947, 48syl 17 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝐽 ∈ Top)
5012clsss3 20786 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑤 𝐽) → ((cls‘𝐽)‘𝑤) ⊆ 𝐽)
5149, 40, 50syl2anc 692 . . . . . . . . . . . . 13 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ 𝐽)
52 f1odm 6103 . . . . . . . . . . . . . 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, 53sseqtr4d 3626 . . . . . . . . . . . 12 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓)
55 funimass3 6294 . . . . . . . . . . . 12 ((Fun 𝑓 ∧ ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
5646, 54, 55syl2anc 692 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
5743, 56mpbird 247 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥)
5842, 57eqsstrd 3623 . . . . . . . . 9 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)
59 eleq2 2687 . . . . . . . . . . 11 (𝑧 = (𝑓𝑤) → (𝑦𝑧𝑦 ∈ (𝑓𝑤)))
60 fveq2 6153 . . . . . . . . . . . 12 (𝑧 = (𝑓𝑤) → ((cls‘𝐾)‘𝑧) = ((cls‘𝐾)‘(𝑓𝑤)))
6160sseq1d 3616 . . . . . . . . . . 11 (𝑧 = (𝑓𝑤) → (((cls‘𝐾)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥))
6259, 61anbi12d 746 . . . . . . . . . 10 (𝑧 = (𝑓𝑤) → ((𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ∈ (𝑓𝑤) ∧ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)))
6362rspcev 3298 . . . . . . . . 9 (((𝑓𝑤) ∈ 𝐾 ∧ (𝑦 ∈ (𝑓𝑤) ∧ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6429, 38, 58, 63syl12anc 1321 . . . . . . . 8 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6525, 64rexlimddv 3029 . . . . . . 7 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6665ralrimivva 2966 . . . . . 6 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → ∀𝑥𝐾𝑦𝑥𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
67 isreg 21059 . . . . . 6 (𝐾 ∈ Reg ↔ (𝐾 ∈ Top ∧ ∀𝑥𝐾𝑦𝑥𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥)))
686, 66, 67sylanbrc 697 . . . . 5 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Reg)
6968expcom 451 . . . 4 (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
7069exlimiv 1855 . . 3 (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
712, 70sylbi 207 . 2 ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
721, 71sylbi 207 1 (𝐽𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  wss 3559  c0 3896   cuni 4407   class class class wbr 4618  ccnv 5078  dom cdm 5079  cima 5082  Fun wfun 5846   Fn wfn 5847  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  Topctop 20630  clsccl 20745   Cn ccn 20951  Regcreg 21036  Homeochmeo 21479  chmph 21480
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-suc 5693  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-1st 7120  df-2nd 7121  df-1o 7512  df-map 7811  df-top 20631  df-topon 20648  df-cld 20746  df-cls 20748  df-cn 20954  df-reg 21043  df-hmeo 21481  df-hmph 21482
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator