This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem homarcl2

Description: Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypotheses homahom.h 
|- H = ( HomA ` C )
homarcl2.b 
|- B = ( Base ` C )
Assertion homarcl2 
|- ( F e. ( X H Y ) -> ( X e. B /\ Y e. B ) )

Proof

Step Hyp Ref Expression
1 homahom.h 
 |-  H = ( HomA ` C )
2 homarcl2.b 
 |-  B = ( Base ` C )
3 elfvdm 
 |-  ( F e. ( H ` <. X , Y >. ) -> <. X , Y >. e. dom H )
4 df-ov 
 |-  ( X H Y ) = ( H ` <. X , Y >. )
5 3 4 eleq2s 
 |-  ( F e. ( X H Y ) -> <. X , Y >. e. dom H )
6 1 homarcl 
 |-  ( F e. ( X H Y ) -> C e. Cat )
7 1 2 6 homaf 
 |-  ( F e. ( X H Y ) -> H : ( B X. B ) --> ~P ( ( B X. B ) X. _V ) )
8 7 fdmd 
 |-  ( F e. ( X H Y ) -> dom H = ( B X. B ) )
9 5 8 eleqtrd 
 |-  ( F e. ( X H Y ) -> <. X , Y >. e. ( B X. B ) )
10 opelxp 
 |-  ( <. X , Y >. e. ( B X. B ) <-> ( X e. B /\ Y e. B ) )
11 9 10 sylib 
 |-  ( F e. ( X H Y ) -> ( X e. B /\ Y e. B ) )