Metamath Proof Explorer


Definition df-retr

Description: Define the set of retractions on two topological spaces. We say that R is a retraction from J to K . or R e. ( J Retr K ) iff there is an S such that R : J --> K , S : K --> J are continuous functions called the retraction and section respectively, and their composite R o. S is homotopic to the identity map. If a retraction exists, we say J is a retract of K . (This terminology is borrowed from HoTT and appears to be nonstandard, although it has similaries to the concept of retract in the category of topological spaces and to a deformation retract in general topology.) Two topological spaces that are retracts of each other are called homotopy equivalent. (Contributed by Mario Carneiro, 11-Feb-2015)

Ref Expression
Assertion df-retr Retr = ( 𝑗 ∈ Top , 𝑘 ∈ Top ↦ { 𝑟 ∈ ( 𝑗 Cn 𝑘 ) ∣ ∃ 𝑠 ∈ ( 𝑘 Cn 𝑗 ) ( ( 𝑟𝑠 ) ( 𝑗 Htpy 𝑗 ) ( I ↾ 𝑗 ) ) ≠ ∅ } )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cretr Retr
1 vj 𝑗
2 ctop Top
3 vk 𝑘
4 vr 𝑟
5 1 cv 𝑗
6 ccn Cn
7 3 cv 𝑘
8 5 7 6 co ( 𝑗 Cn 𝑘 )
9 vs 𝑠
10 7 5 6 co ( 𝑘 Cn 𝑗 )
11 4 cv 𝑟
12 9 cv 𝑠
13 11 12 ccom ( 𝑟𝑠 )
14 chtpy Htpy
15 5 5 14 co ( 𝑗 Htpy 𝑗 )
16 cid I
17 5 cuni 𝑗
18 16 17 cres ( I ↾ 𝑗 )
19 13 18 15 co ( ( 𝑟𝑠 ) ( 𝑗 Htpy 𝑗 ) ( I ↾ 𝑗 ) )
20 c0
21 19 20 wne ( ( 𝑟𝑠 ) ( 𝑗 Htpy 𝑗 ) ( I ↾ 𝑗 ) ) ≠ ∅
22 21 9 10 wrex 𝑠 ∈ ( 𝑘 Cn 𝑗 ) ( ( 𝑟𝑠 ) ( 𝑗 Htpy 𝑗 ) ( I ↾ 𝑗 ) ) ≠ ∅
23 22 4 8 crab { 𝑟 ∈ ( 𝑗 Cn 𝑘 ) ∣ ∃ 𝑠 ∈ ( 𝑘 Cn 𝑗 ) ( ( 𝑟𝑠 ) ( 𝑗 Htpy 𝑗 ) ( I ↾ 𝑗 ) ) ≠ ∅ }
24 1 3 2 2 23 cmpo ( 𝑗 ∈ Top , 𝑘 ∈ Top ↦ { 𝑟 ∈ ( 𝑗 Cn 𝑘 ) ∣ ∃ 𝑠 ∈ ( 𝑘 Cn 𝑗 ) ( ( 𝑟𝑠 ) ( 𝑗 Htpy 𝑗 ) ( I ↾ 𝑗 ) ) ≠ ∅ } )
25 0 24 wceq Retr = ( 𝑗 ∈ Top , 𝑘 ∈ Top ↦ { 𝑟 ∈ ( 𝑗 Cn 𝑘 ) ∣ ∃ 𝑠 ∈ ( 𝑘 Cn 𝑗 ) ( ( 𝑟𝑠 ) ( 𝑗 Htpy 𝑗 ) ( I ↾ 𝑗 ) ) ≠ ∅ } )