Definition df-pi1 25309

Description: Define the fundamental group, whose operation is given by concatenation of homotopy classes of loops. Definition of [Hatcher] p. 26. (Contributed by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
df-pi1	⊢ π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)))

Distinct variable group:   𝑦,𝑗

Detailed syntax breakdown of Definition df-pi1

Step	Hyp	Ref	Expression
1		cpi1 25304	. 2 class π1
2		vj	. . 3 setvar 𝑗
3		vy	. . 3 setvar 𝑦
4		ctop 23191	. . 3 class Top
5	2	cv 1569	. . . 4 class 𝑗
6	5	cuni 4867	. . 3 class ∪ 𝑗
7	3	cv 1569	. . . . 5 class 𝑦
8		comi 25302	. . . . 5 class Ω1
9	5, 7, 8	co 7412	. . . 4 class (𝑗 Ω1 𝑦)
10		cphtpc 25270	. . . . 5 class ≃ph
11	5, 10	cfv 6531	. . . 4 class ( ≃ph‘𝑗)
12		cqus 17657	. . . 4 class /s
13	9, 11, 12	co 7412	. . 3 class ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗))
14	2, 3, 4, 6, 13	cmpo 7414	. 2 class (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)))
15	1, 14	wceq 1570	1 wff π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)))

This definition is used by:  pi1val  25338