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Definition df-inv 16340
Description: The inverse relation in a category. Given arrows 𝑓:𝑋𝑌 and 𝑔:𝑌𝑋, 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
StepHypRef Expression
1 cinv 16337 . 2 class Inv
2 vc . . 3 setvar 𝑐
3 ccat 16257 . . 3 class Cat
4 vx . . . 4 setvar 𝑥
5 vy . . . 4 setvar 𝑦
62cv 1479 . . . . 5 class 𝑐
7 cbs 15792 . . . . 5 class Base
86, 7cfv 5852 . . . 4 class (Base‘𝑐)
94cv 1479 . . . . . 6 class 𝑥
105cv 1479 . . . . . 6 class 𝑦
11 csect 16336 . . . . . . 7 class Sect
126, 11cfv 5852 . . . . . 6 class (Sect‘𝑐)
139, 10, 12co 6610 . . . . 5 class (𝑥(Sect‘𝑐)𝑦)
1410, 9, 12co 6610 . . . . . 6 class (𝑦(Sect‘𝑐)𝑥)
1514ccnv 5078 . . . . 5 class (𝑦(Sect‘𝑐)𝑥)
1613, 15cin 3558 . . . 4 class ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))
174, 5, 8, 8, 16cmpt2 6612 . . 3 class (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥)))
182, 3, 17cmpt 4678 . 2 class (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))))
191, 18wceq 1480 1 wff Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))))
Colors of variables: wff setvar class
This definition is referenced by:  invffval  16350  isofn  16367
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