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Definition df-inv 13667
Description: The inverse relation in a category. Given arrows  f : X --> Y and  g : Y --> X, we say  gInv f, that is,  g is an inverse of  f, if  g is a section of  f and  f is a section of  g. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
df-inv  |- Inv  =  ( c  e.  Cat  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( ( x (Sect `  c )
y )  i^i  `' ( y (Sect `  c ) x ) ) ) )
Distinct variable group:    x, c, y

Detailed syntax breakdown of Definition df-inv
StepHypRef Expression
1 cinv 13664 . 2  class Inv
2 vc . . 3  set  c
3 ccat 13582 . . 3  class  Cat
4 vx . . . 4  set  x
5 vy . . . 4  set  y
62cv 1631 . . . . 5  class  c
7 cbs 13164 . . . . 5  class  Base
86, 7cfv 5271 . . . 4  class  ( Base `  c )
94cv 1631 . . . . . 6  class  x
105cv 1631 . . . . . 6  class  y
11 csect 13663 . . . . . . 7  class Sect
126, 11cfv 5271 . . . . . 6  class  (Sect `  c )
139, 10, 12co 5874 . . . . 5  class  ( x (Sect `  c )
y )
1410, 9, 12co 5874 . . . . . 6  class  ( y (Sect `  c )
x )
1514ccnv 4704 . . . . 5  class  `' ( y (Sect `  c
) x )
1613, 15cin 3164 . . . 4  class  ( ( x (Sect `  c
) y )  i^i  `' ( y (Sect `  c ) x ) )
174, 5, 8, 8, 16cmpt2 5876 . . 3  class  ( x  e.  ( Base `  c
) ,  y  e.  ( Base `  c
)  |->  ( ( x (Sect `  c )
y )  i^i  `' ( y (Sect `  c ) x ) ) )
182, 3, 17cmpt 4093 . 2  class  ( c  e.  Cat  |->  ( x  e.  ( Base `  c
) ,  y  e.  ( Base `  c
)  |->  ( ( x (Sect `  c )
y )  i^i  `' ( y (Sect `  c ) x ) ) ) )
191, 18wceq 1632 1  wff Inv  =  ( c  e.  Cat  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( ( x (Sect `  c )
y )  i^i  `' ( y (Sect `  c ) x ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  invffval  13676
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