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Metamath Proof Explorer

Theorem invf1o

Description: The inverse relation is a bijection from isomorphisms to isomorphisms. This means that every isomorphism F e. ( X I Y ) has a unique inverse, denoted by ( ( InvC )F ) . Remark 3.12 of Adamek p. 28. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypotheses invfval.b 
|- B = ( Base ` C )
invfval.n 
|- N = ( Inv ` C )
invfval.c 
|- ( ph -> C e. Cat )
invss.x 
|- ( ph -> X e. B )
invss.y 
|- ( ph -> Y e. B )
isoval.n 
|- I = ( Iso ` C )
Assertion invf1o 
|- ( ph -> ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) )

Proof

Step Hyp Ref Expression
1 invfval.b 
 |-  B = ( Base ` C )
2 invfval.n 
 |-  N = ( Inv ` C )
3 invfval.c 
 |-  ( ph -> C e. Cat )
4 invss.x 
 |-  ( ph -> X e. B )
5 invss.y 
 |-  ( ph -> Y e. B )
6 isoval.n 
 |-  I = ( Iso ` C )
7 1 2 3 4 5 6 invf 
 |-  ( ph -> ( X N Y ) : ( X I Y ) --> ( Y I X ) )
8 7 ffnd 
 |-  ( ph -> ( X N Y ) Fn ( X I Y ) )
9 1 2 3 5 4 6 invf 
 |-  ( ph -> ( Y N X ) : ( Y I X ) --> ( X I Y ) )
10 9 ffnd 
 |-  ( ph -> ( Y N X ) Fn ( Y I X ) )
11 1 2 3 4 5 invsym2 
 |-  ( ph -> `' ( X N Y ) = ( Y N X ) )
12 11 fneq1d 
 |-  ( ph -> ( `' ( X N Y ) Fn ( Y I X ) <-> ( Y N X ) Fn ( Y I X ) ) )
13 10 12 mpbird 
 |-  ( ph -> `' ( X N Y ) Fn ( Y I X ) )
14 dff1o4 
 |-  ( ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) <-> ( ( X N Y ) Fn ( X I Y ) /\ `' ( X N Y ) Fn ( Y I X ) ) )
15 8 13 14 sylanbrc 
 |-  ( ph -> ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) )