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Metamath Proof Explorer

Definition df-inito

Description: An object A is said to be an initial object provided that for each object B there is exactly one morphism from A to B. Definition 7.1 in Adamek p. 101, or definition in Lang p. 57 (called "a universally repelling object" there). See dfinito2 and dfinito3 for alternate definitions depending on df-termo . See dfinito4 for an alternate definition using the universal property. (Contributed by AV, 3-Apr-2020)

Ref Expression
Assertion df-inito 
|- InitO = ( c e. Cat |-> { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) } )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cinito 
 |-  InitO
1 vc 
 |-  c
2 ccat 
 |-  Cat
3 va 
 |-  a
4 cbs 
 |-  Base
5 1 cv 
 |-  c
6 5 4 cfv 
 |-  ( Base ` c )
7 vb 
 |-  b
8 vh 
 |-  h
9 8 cv 
 |-  h
10 3 cv 
 |-  a
11 chom 
 |-  Hom
12 5 11 cfv 
 |-  ( Hom ` c )
13 7 cv 
 |-  b
14 10 13 12 co 
 |-  ( a ( Hom ` c ) b )
15 9 14 wcel 
 |-  h e. ( a ( Hom ` c ) b )
16 15 8 weu 
 |-  E! h h e. ( a ( Hom ` c ) b )
17 16 7 6 wral 
 |-  A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b )
18 17 3 6 crab 
 |-  { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) }
19 1 2 18 cmpt 
 |-  ( c e. Cat |-> { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) } )
20 0 19 wceq 
 |-  InitO = ( c e. Cat |-> { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) } )