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	sseldi	⊢ ( 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 )