Metamath Proof Explorer


Definition df-pconn

Description: Define the class of path-connected topologies. A topology is path-connected if there is a path (a continuous function from the closed unit interval) that goes from x to y for any points x , y in the space. (Contributed by Mario Carneiro, 11-Feb-2015)

Ref Expression
Assertion df-pconn PConn = { 𝑗 ∈ Top ∣ ∀ 𝑥 𝑗𝑦 𝑗𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cpconn PConn
1 vj 𝑗
2 ctop Top
3 vx 𝑥
4 1 cv 𝑗
5 4 cuni 𝑗
6 vy 𝑦
7 vf 𝑓
8 cii II
9 ccn Cn
10 8 4 9 co ( II Cn 𝑗 )
11 7 cv 𝑓
12 cc0 0
13 12 11 cfv ( 𝑓 ‘ 0 )
14 3 cv 𝑥
15 13 14 wceq ( 𝑓 ‘ 0 ) = 𝑥
16 c1 1
17 16 11 cfv ( 𝑓 ‘ 1 )
18 6 cv 𝑦
19 17 18 wceq ( 𝑓 ‘ 1 ) = 𝑦
20 15 19 wa ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 )
21 20 7 10 wrex 𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 )
22 21 6 5 wral 𝑦 𝑗𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 )
23 22 3 5 wral 𝑥 𝑗𝑦 𝑗𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 )
24 23 1 2 crab { 𝑗 ∈ Top ∣ ∀ 𝑥 𝑗𝑦 𝑗𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) }
25 0 24 wceq PConn = { 𝑗 ∈ Top ∣ ∀ 𝑥 𝑗𝑦 𝑗𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) }