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Metamath Proof Explorer

Theorem resscat

Description: A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017)

Ref Expression
Assertion resscat 
|- ( ( C e. Cat /\ S e. V ) -> ( C |`s S ) e. Cat )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Base ` C ) = ( Base ` C )
2 1 ressinbas 
 |-  ( S e. V -> ( C |`s S ) = ( C |`s ( S i^i ( Base ` C ) ) ) )
3 2 adantl 
 |-  ( ( C e. Cat /\ S e. V ) -> ( C |`s S ) = ( C |`s ( S i^i ( Base ` C ) ) ) )
4 eqid 
 |-  ( C |`cat ( ( Homf ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) = ( C |`cat ( ( Homf ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) )
5 eqid 
 |-  ( Homf ` C ) = ( Homf ` C )
6 simpl 
 |-  ( ( C e. Cat /\ S e. V ) -> C e. Cat )
7 inss2 
 |-  ( S i^i ( Base ` C ) ) C_ ( Base ` C )
8 7 a1i 
 |-  ( ( C e. Cat /\ S e. V ) -> ( S i^i ( Base ` C ) ) C_ ( Base ` C ) )
9 1 5 6 8 fullsubc 
 |-  ( ( C e. Cat /\ S e. V ) -> ( ( Homf ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) e. ( Subcat ` C ) )
10 4 9 subccat 
 |-  ( ( C e. Cat /\ S e. V ) -> ( C |`cat ( ( Homf ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) e. Cat )
11 eqid 
 |-  ( C |`s ( S i^i ( Base ` C ) ) ) = ( C |`s ( S i^i ( Base ` C ) ) )
12 1 5 6 8 11 4 fullresc 
 |-  ( ( C e. Cat /\ S e. V ) -> ( ( Homf ` ( C |`s ( S i^i ( Base ` C ) ) ) ) = ( Homf ` ( C |`cat ( ( Homf ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) /\ ( comf ` ( C |`s ( S i^i ( Base ` C ) ) ) ) = ( comf ` ( C |`cat ( ( Homf ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) ) )
13 12 simpld 
 |-  ( ( C e. Cat /\ S e. V ) -> ( Homf ` ( C |`s ( S i^i ( Base ` C ) ) ) ) = ( Homf ` ( C |`cat ( ( Homf ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) )
14 12 simprd 
 |-  ( ( C e. Cat /\ S e. V ) -> ( comf ` ( C |`s ( S i^i ( Base ` C ) ) ) ) = ( comf ` ( C |`cat ( ( Homf ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) )
15 ovexd 
 |-  ( ( C e. Cat /\ S e. V ) -> ( C |`s ( S i^i ( Base ` C ) ) ) e. _V )
16 13 14 15 10 catpropd 
 |-  ( ( C e. Cat /\ S e. V ) -> ( ( C |`s ( S i^i ( Base ` C ) ) ) e. Cat <-> ( C |`cat ( ( Homf ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) e. Cat ) )
17 10 16 mpbird 
 |-  ( ( C e. Cat /\ S e. V ) -> ( C |`s ( S i^i ( Base ` C ) ) ) e. Cat )
18 3 17 eqeltrd 
 |-  ( ( C e. Cat /\ S e. V ) -> ( C |`s S ) e. Cat )