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Metamath Proof Explorer

Theorem istermo

Description: The predicate "is a terminal object" of a category. (Contributed by AV, 3-Apr-2020)

Ref Expression
Hypotheses isinito.b 
|- B = ( Base ` C )
isinito.h 
|- H = ( Hom ` C )
isinito.c 
|- ( ph -> C e. Cat )
isinito.i 
|- ( ph -> I e. B )
Assertion istermo 
|- ( ph -> ( I e. ( TermO ` C ) <-> A. b e. B E! h h e. ( b H I ) ) )

Proof

Step Hyp Ref Expression
1 isinito.b 
 |-  B = ( Base ` C )
2 isinito.h 
 |-  H = ( Hom ` C )
3 isinito.c 
 |-  ( ph -> C e. Cat )
4 isinito.i 
 |-  ( ph -> I e. B )
5 3 1 2 termoval 
 |-  ( ph -> ( TermO ` C ) = { i e. B | A. b e. B E! h h e. ( b H i ) } )
6 5 eleq2d 
 |-  ( ph -> ( I e. ( TermO ` C ) <-> I e. { i e. B | A. b e. B E! h h e. ( b H i ) } ) )
7 oveq2 
 |-  ( i = I -> ( b H i ) = ( b H I ) )
8 7 eleq2d 
 |-  ( i = I -> ( h e. ( b H i ) <-> h e. ( b H I ) ) )
9 8 eubidv 
 |-  ( i = I -> ( E! h h e. ( b H i ) <-> E! h h e. ( b H I ) ) )
10 9 ralbidv 
 |-  ( i = I -> ( A. b e. B E! h h e. ( b H i ) <-> A. b e. B E! h h e. ( b H I ) ) )
11 10 elrab3 
 |-  ( I e. B -> ( I e. { i e. B | A. b e. B E! h h e. ( b H i ) } <-> A. b e. B E! h h e. ( b H I ) ) )
12 4 11 syl 
 |-  ( ph -> ( I e. { i e. B | A. b e. B E! h h e. ( b H i ) } <-> A. b e. B E! h h e. ( b H I ) ) )
13 6 12 bitrd 
 |-  ( ph -> ( I e. ( TermO ` C ) <-> A. b e. B E! h h e. ( b H I ) ) )