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 . (Contributed by AV, 3-Apr-2020)

Ref Expression
Assertion df-inito InitO = ( 𝑐 ∈ Cat ↦ { 𝑎 ∈ ( Base ‘ 𝑐 ) ∣ ∀ 𝑏 ∈ ( Base ‘ 𝑐 ) ∃! ∈ ( 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
9 8 cv
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 ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 )
16 15 8 weu ∃! ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 )
17 16 7 6 wral 𝑏 ∈ ( Base ‘ 𝑐 ) ∃! ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 )
18 17 3 6 crab { 𝑎 ∈ ( Base ‘ 𝑐 ) ∣ ∀ 𝑏 ∈ ( Base ‘ 𝑐 ) ∃! ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 ) }
19 1 2 18 cmpt ( 𝑐 ∈ Cat ↦ { 𝑎 ∈ ( Base ‘ 𝑐 ) ∣ ∀ 𝑏 ∈ ( Base ‘ 𝑐 ) ∃! ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 ) } )
20 0 19 wceq InitO = ( 𝑐 ∈ Cat ↦ { 𝑎 ∈ ( Base ‘ 𝑐 ) ∣ ∀ 𝑏 ∈ ( Base ‘ 𝑐 ) ∃! ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 ) } )