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Metamath Proof Explorer

Theorem catrid

Description: Right 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 ⊢ ( 𝜑 → 𝐶 ∈ Cat )
catidcl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
catlid.o ⊢ · = ( comp ‘ 𝐶 )
catlid.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
catlid.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
Assertion catrid ( 𝜑 → ( 𝐹 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 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 ⊢ ( 𝜑 → 𝐶 ∈ Cat )
5 catidcl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 catlid.o ⊢ · = ( comp ‘ 𝐶 )
7 catlid.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
8 catlid.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
9 oveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑌 ) ( 1 ‘ 𝑋 ) ) = ( 𝐹 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑌 ) ( 1 ‘ 𝑋 ) ) )
10 id ⊢ ( 𝑓 = 𝐹 → 𝑓 = 𝐹 )
11 9 10 eqeq12d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ↔ ( 𝐹 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 𝐹 ) )
12 oveq2 ⊢ ( 𝑦 = 𝑌 → ( 𝑋 𝐻 𝑦 ) = ( 𝑋 𝐻 𝑌 ) )
13 oveq2 ⊢ ( 𝑦 = 𝑌 → ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) = ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑌 ) )
14 13 oveqd ⊢ ( 𝑦 = 𝑌 → ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) ( 1 ‘ 𝑋 ) ) = ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑌 ) ( 1 ‘ 𝑋 ) ) )
15 14 eqeq1d ⊢ ( 𝑦 = 𝑌 → ( ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ↔ ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) )
16 12 15 raleqbidv ⊢ ( 𝑦 = 𝑌 → ( ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ↔ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) )
17 simpr ⊢ ( ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) → ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 )
18 17 ralimi ⊢ ( ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) → ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 )
19 18 a1i ⊢ ( 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) → ( ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) → ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) )
20 19 ss2rabi ⊢ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) } ⊆ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 }
21 1 2 6 4 3 5 cidval ⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) = ( ℩ 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) )
22 1 2 6 4 5 catideu ⊢ ( 𝜑 → ∃! 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) )
23 riotacl2 ⊢ ( ∃! 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) → ( ℩ 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) ∈ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) } )
24 22 23 syl ⊢ ( 𝜑 → ( ℩ 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) ∈ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) } )
25 21 24 eqeltrd ⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) ∈ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) } )
26 20 25 sselid ⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) ∈ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 } )
27 oveq2 ⊢ ( 𝑔 = ( 1 ‘ 𝑋 ) → ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) ( 1 ‘ 𝑋 ) ) )
28 27 eqeq1d ⊢ ( 𝑔 = ( 1 ‘ 𝑋 ) → ( ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ↔ ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) )
29 28 2ralbidv ⊢ ( 𝑔 = ( 1 ‘ 𝑋 ) → ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ↔ ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) )
30 29 elrab ⊢ ( ( 1 ‘ 𝑋 ) ∈ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 } ↔ ( ( 1 ‘ 𝑋 ) ∈ ( 𝑋 𝐻 𝑋 ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) )
31 30 simprbi ⊢ ( ( 1 ‘ 𝑋 ) ∈ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 } → ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 )
32 26 31 syl ⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 )
33 16 32 7 rspcdva ⊢ ( 𝜑 → ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 𝑓 )
34 11 33 8 rspcdva ⊢ ( 𝜑 → ( 𝐹 ( ⟨ 𝑋 , 𝑋 ⟩ · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 𝐹 )