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Metamath Proof Explorer

Theorem homarel

Description: An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypothesis homahom.h 
|- H = ( HomA ` C )
Assertion homarel 
|- Rel ( X H Y )

Proof

Step Hyp Ref Expression
1 homahom.h 
 |-  H = ( HomA ` C )
2 xpss 
 |-  ( ( ( Base ` C ) X. ( Base ` C ) ) X. _V ) C_ ( _V X. _V )
3 eqid 
 |-  ( Base ` C ) = ( Base ` C )
4 1 homarcl 
 |-  ( f e. ( X H Y ) -> C e. Cat )
5 1 3 4 homaf 
 |-  ( f e. ( X H Y ) -> H : ( ( Base ` C ) X. ( Base ` C ) ) --> ~P ( ( ( Base ` C ) X. ( Base ` C ) ) X. _V ) )
6 1 3 homarcl2 
 |-  ( f e. ( X H Y ) -> ( X e. ( Base ` C ) /\ Y e. ( Base ` C ) ) )
7 6 simpld 
 |-  ( f e. ( X H Y ) -> X e. ( Base ` C ) )
8 6 simprd 
 |-  ( f e. ( X H Y ) -> Y e. ( Base ` C ) )
9 5 7 8 fovcdmd 
 |-  ( f e. ( X H Y ) -> ( X H Y ) e. ~P ( ( ( Base ` C ) X. ( Base ` C ) ) X. _V ) )
10 elelpwi 
 |-  ( ( f e. ( X H Y ) /\ ( X H Y ) e. ~P ( ( ( Base ` C ) X. ( Base ` C ) ) X. _V ) ) -> f e. ( ( ( Base ` C ) X. ( Base ` C ) ) X. _V ) )
11 9 10 mpdan 
 |-  ( f e. ( X H Y ) -> f e. ( ( ( Base ` C ) X. ( Base ` C ) ) X. _V ) )
12 2 11 sselid 
 |-  ( f e. ( X H Y ) -> f e. ( _V X. _V ) )
13 12 ssriv 
 |-  ( X H Y ) C_ ( _V X. _V )
14 df-rel 
 |-  ( Rel ( X H Y ) <-> ( X H Y ) C_ ( _V X. _V ) )
15 13 14 mpbir 
 |-  Rel ( X H Y )