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Metamath Proof Explorer

Theorem elhomai2

Description: Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypotheses homarcl.h 
|- H = ( HomA ` C )
homafval.b 
|- B = ( Base ` C )
homafval.c 
|- ( ph -> C e. Cat )
homaval.j 
|- J = ( Hom ` C )
homaval.x 
|- ( ph -> X e. B )
homaval.y 
|- ( ph -> Y e. B )
elhomai.f 
|- ( ph -> F e. ( X J Y ) )
Assertion elhomai2 
|- ( ph -> <. X , Y , F >. e. ( X H Y ) )

Proof

Step Hyp Ref Expression
1 homarcl.h 
 |-  H = ( HomA ` C )
2 homafval.b 
 |-  B = ( Base ` C )
3 homafval.c 
 |-  ( ph -> C e. Cat )
4 homaval.j 
 |-  J = ( Hom ` C )
5 homaval.x 
 |-  ( ph -> X e. B )
6 homaval.y 
 |-  ( ph -> Y e. B )
7 elhomai.f 
 |-  ( ph -> F e. ( X J Y ) )
8 df-ot 
 |-  <. X , Y , F >. = <. <. X , Y >. , F >.
9 1 2 3 4 5 6 7 elhomai 
 |-  ( ph -> <. X , Y >. ( X H Y ) F )
10 df-br 
 |-  ( <. X , Y >. ( X H Y ) F <-> <. <. X , Y >. , F >. e. ( X H Y ) )
11 9 10 sylib 
 |-  ( ph -> <. <. X , Y >. , F >. e. ( X H Y ) )
12 8 11 eqeltrid 
 |-  ( ph -> <. X , Y , F >. e. ( X H Y ) )