# Metamath Proof Explorer

## Theorem catlid

Description: Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypotheses catidcl.b 𝐵 = ( Base ‘ 𝐶 )
catidcl.h 𝐻 = ( Hom ‘ 𝐶 )
catidcl.i 1 = ( Id ‘ 𝐶 )
catidcl.c ( 𝜑𝐶 ∈ Cat )
catidcl.x ( 𝜑𝑋𝐵 )
catlid.o · = ( comp ‘ 𝐶 )
catlid.y ( 𝜑𝑌𝐵 )
catlid.f ( 𝜑𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
Assertion catlid ( 𝜑 → ( ( 1𝑌 ) ( ⟨ 𝑋 , 𝑌· 𝑌 ) 𝐹 ) = 𝐹 )

### Proof

Step Hyp Ref Expression
1 catidcl.b 𝐵 = ( Base ‘ 𝐶 )
2 catidcl.h 𝐻 = ( Hom ‘ 𝐶 )
3 catidcl.i 1 = ( Id ‘ 𝐶 )
4 catidcl.c ( 𝜑𝐶 ∈ Cat )
5 catidcl.x ( 𝜑𝑋𝐵 )
6 catlid.o · = ( comp ‘ 𝐶 )
7 catlid.y ( 𝜑𝑌𝐵 )
8 catlid.f ( 𝜑𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
9 oveq2 ( 𝑓 = 𝐹 → ( ( 1𝑌 ) ( ⟨ 𝑋 , 𝑌· 𝑌 ) 𝑓 ) = ( ( 1𝑌 ) ( ⟨ 𝑋 , 𝑌· 𝑌 ) 𝐹 ) )
10 id ( 𝑓 = 𝐹𝑓 = 𝐹 )
11 9 10 eqeq12d ( 𝑓 = 𝐹 → ( ( ( 1𝑌 ) ( ⟨ 𝑋 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ↔ ( ( 1𝑌 ) ( ⟨ 𝑋 , 𝑌· 𝑌 ) 𝐹 ) = 𝐹 ) )
12 oveq1 ( 𝑥 = 𝑋 → ( 𝑥 𝐻 𝑌 ) = ( 𝑋 𝐻 𝑌 ) )
13 opeq1 ( 𝑥 = 𝑋 → ⟨ 𝑥 , 𝑌 ⟩ = ⟨ 𝑋 , 𝑌 ⟩ )
14 13 oveq1d ( 𝑥 = 𝑋 → ( ⟨ 𝑥 , 𝑌· 𝑌 ) = ( ⟨ 𝑋 , 𝑌· 𝑌 ) )
15 14 oveqd ( 𝑥 = 𝑋 → ( ( 1𝑌 ) ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = ( ( 1𝑌 ) ( ⟨ 𝑋 , 𝑌· 𝑌 ) 𝑓 ) )
16 15 eqeq1d ( 𝑥 = 𝑋 → ( ( ( 1𝑌 ) ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ↔ ( ( 1𝑌 ) ( ⟨ 𝑋 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ) )
17 12 16 raleqbidv ( 𝑥 = 𝑋 → ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( ( 1𝑌 ) ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ↔ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ( ( 1𝑌 ) ( ⟨ 𝑋 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ) )
18 simpl ( ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑌 𝐻 𝑥 ) ( 𝑓 ( ⟨ 𝑌 , 𝑌· 𝑥 ) 𝑔 ) = 𝑓 ) → ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 )
19 18 ralimi ( ∀ 𝑥𝐵 ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑌 𝐻 𝑥 ) ( 𝑓 ( ⟨ 𝑌 , 𝑌· 𝑥 ) 𝑔 ) = 𝑓 ) → ∀ 𝑥𝐵𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 )
20 19 a1i ( 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) → ( ∀ 𝑥𝐵 ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑌 𝐻 𝑥 ) ( 𝑓 ( ⟨ 𝑌 , 𝑌· 𝑥 ) 𝑔 ) = 𝑓 ) → ∀ 𝑥𝐵𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ) )
21 20 ss2rabi { 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∣ ∀ 𝑥𝐵 ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑌 𝐻 𝑥 ) ( 𝑓 ( ⟨ 𝑌 , 𝑌· 𝑥 ) 𝑔 ) = 𝑓 ) } ⊆ { 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∣ ∀ 𝑥𝐵𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 }
22 1 2 6 4 3 7 cidval ( 𝜑 → ( 1𝑌 ) = ( 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∀ 𝑥𝐵 ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑌 𝐻 𝑥 ) ( 𝑓 ( ⟨ 𝑌 , 𝑌· 𝑥 ) 𝑔 ) = 𝑓 ) ) )
23 1 2 6 4 7 catideu ( 𝜑 → ∃! 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∀ 𝑥𝐵 ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑌 𝐻 𝑥 ) ( 𝑓 ( ⟨ 𝑌 , 𝑌· 𝑥 ) 𝑔 ) = 𝑓 ) )
24 riotacl2 ( ∃! 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∀ 𝑥𝐵 ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑌 𝐻 𝑥 ) ( 𝑓 ( ⟨ 𝑌 , 𝑌· 𝑥 ) 𝑔 ) = 𝑓 ) → ( 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∀ 𝑥𝐵 ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑌 𝐻 𝑥 ) ( 𝑓 ( ⟨ 𝑌 , 𝑌· 𝑥 ) 𝑔 ) = 𝑓 ) ) ∈ { 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∣ ∀ 𝑥𝐵 ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑌 𝐻 𝑥 ) ( 𝑓 ( ⟨ 𝑌 , 𝑌· 𝑥 ) 𝑔 ) = 𝑓 ) } )
25 23 24 syl ( 𝜑 → ( 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∀ 𝑥𝐵 ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑌 𝐻 𝑥 ) ( 𝑓 ( ⟨ 𝑌 , 𝑌· 𝑥 ) 𝑔 ) = 𝑓 ) ) ∈ { 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∣ ∀ 𝑥𝐵 ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑌 𝐻 𝑥 ) ( 𝑓 ( ⟨ 𝑌 , 𝑌· 𝑥 ) 𝑔 ) = 𝑓 ) } )
26 22 25 eqeltrd ( 𝜑 → ( 1𝑌 ) ∈ { 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∣ ∀ 𝑥𝐵 ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑌 𝐻 𝑥 ) ( 𝑓 ( ⟨ 𝑌 , 𝑌· 𝑥 ) 𝑔 ) = 𝑓 ) } )
27 21 26 sseldi ( 𝜑 → ( 1𝑌 ) ∈ { 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∣ ∀ 𝑥𝐵𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 } )
28 oveq1 ( 𝑔 = ( 1𝑌 ) → ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = ( ( 1𝑌 ) ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) )
29 28 eqeq1d ( 𝑔 = ( 1𝑌 ) → ( ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ↔ ( ( 1𝑌 ) ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ) )
30 29 2ralbidv ( 𝑔 = ( 1𝑌 ) → ( ∀ 𝑥𝐵𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ↔ ∀ 𝑥𝐵𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( ( 1𝑌 ) ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ) )
31 30 elrab ( ( 1𝑌 ) ∈ { 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∣ ∀ 𝑥𝐵𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 } ↔ ( ( 1𝑌 ) ∈ ( 𝑌 𝐻 𝑌 ) ∧ ∀ 𝑥𝐵𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( ( 1𝑌 ) ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 ) )
32 31 simprbi ( ( 1𝑌 ) ∈ { 𝑔 ∈ ( 𝑌 𝐻 𝑌 ) ∣ ∀ 𝑥𝐵𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( 𝑔 ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 } → ∀ 𝑥𝐵𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( ( 1𝑌 ) ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 )
33 27 32 syl ( 𝜑 → ∀ 𝑥𝐵𝑓 ∈ ( 𝑥 𝐻 𝑌 ) ( ( 1𝑌 ) ( ⟨ 𝑥 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 )
34 17 33 5 rspcdva ( 𝜑 → ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ( ( 1𝑌 ) ( ⟨ 𝑋 , 𝑌· 𝑌 ) 𝑓 ) = 𝑓 )
35 11 34 8 rspcdva ( 𝜑 → ( ( 1𝑌 ) ( ⟨ 𝑋 , 𝑌· 𝑌 ) 𝐹 ) = 𝐹 )