Definition df-inv 17377

Description: The inverse relation in a category. Given arrows 𝑓:𝑋⟶𝑌 and 𝑔:𝑌⟶𝑋, we say 𝑔Inv𝑓, that is, 𝑔 is an inverse of 𝑓, if 𝑔 is a section of 𝑓 and 𝑓 is a section of 𝑔. Definition 3.8 of [Adamek] p. 28. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
df-inv	⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))))

Distinct variable group:   𝑥,𝑐,𝑦

Detailed syntax breakdown of Definition df-inv

Step	Hyp	Ref	Expression
1		cinv 17374	. 2 class Inv
2		vc	. . 3 setvar 𝑐
3		ccat 17290	. . 3 class Cat
4		vx	. . . 4 setvar 𝑥
5		vy	. . . 4 setvar 𝑦
6	2	cv 1538	. . . . 5 class 𝑐
7		cbs 16840	. . . . 5 class Base
8	6, 7	cfv 6418	. . . 4 class (Base‘𝑐)
9	4	cv 1538	. . . . . 6 class 𝑥
10	5	cv 1538	. . . . . 6 class 𝑦
11		csect 17373	. . . . . . 7 class Sect
12	6, 11	cfv 6418	. . . . . 6 class (Sect‘𝑐)
13	9, 10, 12	co 7255	. . . . 5 class (𝑥(Sect‘𝑐)𝑦)
14	10, 9, 12	co 7255	. . . . . 6 class (𝑦(Sect‘𝑐)𝑥)
15	14	ccnv 5579	. . . . 5 class ◡(𝑦(Sect‘𝑐)𝑥)
16	13, 15	cin 3882	. . . 4 class ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))
17	4, 5, 8, 8, 16	cmpo 7257	. . 3 class (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)))
18	2, 3, 17	cmpt 5153	. 2 class (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))))
19	1, 18	wceq 1539	1 wff Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))))

This definition is referenced by:  invffval  17387  isofn  17404