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Mirrors > Home > MPE Home > Th. List > df-inv | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-inv | ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cinv 17255 | . 2 class Inv | |
2 | vc | . . 3 setvar 𝑐 | |
3 | ccat 17172 | . . 3 class Cat | |
4 | vx | . . . 4 setvar 𝑥 | |
5 | vy | . . . 4 setvar 𝑦 | |
6 | 2 | cv 1542 | . . . . 5 class 𝑐 |
7 | cbs 16765 | . . . . 5 class Base | |
8 | 6, 7 | cfv 6385 | . . . 4 class (Base‘𝑐) |
9 | 4 | cv 1542 | . . . . . 6 class 𝑥 |
10 | 5 | cv 1542 | . . . . . 6 class 𝑦 |
11 | csect 17254 | . . . . . . 7 class Sect | |
12 | 6, 11 | cfv 6385 | . . . . . 6 class (Sect‘𝑐) |
13 | 9, 10, 12 | co 7218 | . . . . 5 class (𝑥(Sect‘𝑐)𝑦) |
14 | 10, 9, 12 | co 7218 | . . . . . 6 class (𝑦(Sect‘𝑐)𝑥) |
15 | 14 | ccnv 5555 | . . . . 5 class ◡(𝑦(Sect‘𝑐)𝑥) |
16 | 13, 15 | cin 3870 | . . . 4 class ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)) |
17 | 4, 5, 8, 8, 16 | cmpo 7220 | . . 3 class (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))) |
18 | 2, 3, 17 | cmpt 5140 | . 2 class (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)))) |
19 | 1, 18 | wceq 1543 | 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 17268 isofn 17285 |
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