Metamath Proof Explorer


Theorem cnrest2r

Description: Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Madsen, 2-Sep-2009) (Revised by Mario Carneiro, 7-Jun-2014)

Ref Expression
Assertion cnrest2r K Top J Cn K 𝑡 B J Cn K

Proof

Step Hyp Ref Expression
1 simpr K Top f J Cn K 𝑡 B f J Cn K 𝑡 B
2 cntop2 f J Cn K 𝑡 B K 𝑡 B Top
3 2 adantl K Top f J Cn K 𝑡 B K 𝑡 B Top
4 restrcl K 𝑡 B Top K V B V
5 eqid K = K
6 5 restin K V B V K 𝑡 B = K 𝑡 B K
7 3 4 6 3syl K Top f J Cn K 𝑡 B K 𝑡 B = K 𝑡 B K
8 7 oveq2d K Top f J Cn K 𝑡 B J Cn K 𝑡 B = J Cn K 𝑡 B K
9 1 8 eleqtrd K Top f J Cn K 𝑡 B f J Cn K 𝑡 B K
10 simpl K Top f J Cn K 𝑡 B K Top
11 toptopon2 K Top K TopOn K
12 10 11 sylib K Top f J Cn K 𝑡 B K TopOn K
13 cntop1 f J Cn K 𝑡 B J Top
14 13 adantl K Top f J Cn K 𝑡 B J Top
15 toptopon2 J Top J TopOn J
16 14 15 sylib K Top f J Cn K 𝑡 B J TopOn J
17 inss2 B K K
18 resttopon K TopOn K B K K K 𝑡 B K TopOn B K
19 12 17 18 sylancl K Top f J Cn K 𝑡 B K 𝑡 B K TopOn B K
20 cnf2 J TopOn J K 𝑡 B K TopOn B K f J Cn K 𝑡 B K f : J B K
21 16 19 9 20 syl3anc K Top f J Cn K 𝑡 B f : J B K
22 21 frnd K Top f J Cn K 𝑡 B ran f B K
23 17 a1i K Top f J Cn K 𝑡 B B K K
24 cnrest2 K TopOn K ran f B K B K K f J Cn K f J Cn K 𝑡 B K
25 12 22 23 24 syl3anc K Top f J Cn K 𝑡 B f J Cn K f J Cn K 𝑡 B K
26 9 25 mpbird K Top f J Cn K 𝑡 B f J Cn K
27 26 ex K Top f J Cn K 𝑡 B f J Cn K
28 27 ssrdv K Top J Cn K 𝑡 B J Cn K