diag12

Description: Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017) (Revised by Mario Carneiro, 15-Jan-2017)

	Ref	Expression
Hypotheses	diagval.l	⊢ 𝐿 = ( 𝐶 Δ_func 𝐷 )
	diagval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	diagval.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
	diag11.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
	diag11.c	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
	diag11.k	⊢ 𝐾 = ( ( 1^st ‘ 𝐿 ) ‘ 𝑋 )
	diag11.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
	diag11.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	diag12.j	⊢ 𝐽 = ( Hom ‘ 𝐷 )
	diag12.i	⊢ 1 = ( Id ‘ 𝐶 )
	diag12.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
	diag12.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑌 𝐽 𝑍 ) )
Assertion	diag12	⊢ ( 𝜑 → ( ( 𝑌 ( 2^nd ‘ 𝐾 ) 𝑍 ) ‘ 𝐹 ) = ( 1 ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		diagval.l	⊢ 𝐿 = ( 𝐶 Δ_func 𝐷 )
2		diagval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
3		diagval.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
4		diag11.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
5		diag11.c	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
6		diag11.k	⊢ 𝐾 = ( ( 1^st ‘ 𝐿 ) ‘ 𝑋 )
7		diag11.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
8		diag11.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
9		diag12.j	⊢ 𝐽 = ( Hom ‘ 𝐷 )
10		diag12.i	⊢ 1 = ( Id ‘ 𝐶 )
11		diag12.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
12		diag12.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑌 𝐽 𝑍 ) )
13	1 2 3	diagval	⊢ ( 𝜑 → 𝐿 = ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) )
14	13	fveq2d	⊢ ( 𝜑 → ( 1^st ‘ 𝐿 ) = ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) )
15	14	fveq1d	⊢ ( 𝜑 → ( ( 1^st ‘ 𝐿 ) ‘ 𝑋 ) = ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) ‘ 𝑋 ) )
16	6 15	eqtrid	⊢ ( 𝜑 → 𝐾 = ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) ‘ 𝑋 ) )
17	16	fveq2d	⊢ ( 𝜑 → ( 2^nd ‘ 𝐾 ) = ( 2^nd ‘ ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) ‘ 𝑋 ) ) )
18	17	oveqd	⊢ ( 𝜑 → ( 𝑌 ( 2^nd ‘ 𝐾 ) 𝑍 ) = ( 𝑌 ( 2^nd ‘ ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) ‘ 𝑋 ) ) 𝑍 ) )
19	18	fveq1d	⊢ ( 𝜑 → ( ( 𝑌 ( 2^nd ‘ 𝐾 ) 𝑍 ) ‘ 𝐹 ) = ( ( 𝑌 ( 2^nd ‘ ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) ‘ 𝑋 ) ) 𝑍 ) ‘ 𝐹 ) )
20		eqid	⊢ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) = ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) )
21		eqid	⊢ ( 𝐶 ×_c 𝐷 ) = ( 𝐶 ×_c 𝐷 )
22		eqid	⊢ ( 𝐶 1^st_F 𝐷 ) = ( 𝐶 1^st_F 𝐷 )
23	21 2 3 22	1stfcl	⊢ ( 𝜑 → ( 𝐶 1^st_F 𝐷 ) ∈ ( ( 𝐶 ×_c 𝐷 ) Func 𝐶 ) )
24		eqid	⊢ ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) ‘ 𝑋 ) = ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) ‘ 𝑋 )
25	20 4 2 3 23 7 5 24 8 9 10 11 12	curf12	⊢ ( 𝜑 → ( ( 𝑌 ( 2^nd ‘ ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) ‘ 𝑋 ) ) 𝑍 ) ‘ 𝐹 ) = ( ( 1 ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑌 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) 𝐹 ) )
26		df-ov	⊢ ( ( 1 ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑌 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) 𝐹 ) = ( ( ⟨ 𝑋 , 𝑌 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) ‘ ⟨ ( 1 ‘ 𝑋 ) , 𝐹 ⟩ )
27	21 4 7	xpcbas	⊢ ( 𝐴 × 𝐵 ) = ( Base ‘ ( 𝐶 ×_c 𝐷 ) )
28		eqid	⊢ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) = ( Hom ‘ ( 𝐶 ×_c 𝐷 ) )
29	5 8	opelxpd	⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐴 × 𝐵 ) )
30	5 11	opelxpd	⊢ ( 𝜑 → ⟨ 𝑋 , 𝑍 ⟩ ∈ ( 𝐴 × 𝐵 ) )
31	21 27 28 2 3 22 29 30	1stf2	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) = ( 1^st ↾ ( ⟨ 𝑋 , 𝑌 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) ) )
32	31	fveq1d	⊢ ( 𝜑 → ( ( ⟨ 𝑋 , 𝑌 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) ‘ ⟨ ( 1 ‘ 𝑋 ) , 𝐹 ⟩ ) = ( ( 1^st ↾ ( ⟨ 𝑋 , 𝑌 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) ) ‘ ⟨ ( 1 ‘ 𝑋 ) , 𝐹 ⟩ ) )
33	26 32	eqtrid	⊢ ( 𝜑 → ( ( 1 ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑌 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) 𝐹 ) = ( ( 1^st ↾ ( ⟨ 𝑋 , 𝑌 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) ) ‘ ⟨ ( 1 ‘ 𝑋 ) , 𝐹 ⟩ ) )
34		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
35	4 34 10 2 5	catidcl	⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) )
36	35 12	opelxpd	⊢ ( 𝜑 → ⟨ ( 1 ‘ 𝑋 ) , 𝐹 ⟩ ∈ ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑌 𝐽 𝑍 ) ) )
37	21 4 7 34 9 5 8 5 11 28	xpchom2	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) = ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑌 𝐽 𝑍 ) ) )
38	36 37	eleqtrrd	⊢ ( 𝜑 → ⟨ ( 1 ‘ 𝑋 ) , 𝐹 ⟩ ∈ ( ⟨ 𝑋 , 𝑌 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) )
39	38	fvresd	⊢ ( 𝜑 → ( ( 1^st ↾ ( ⟨ 𝑋 , 𝑌 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) ) ‘ ⟨ ( 1 ‘ 𝑋 ) , 𝐹 ⟩ ) = ( 1^st ‘ ⟨ ( 1 ‘ 𝑋 ) , 𝐹 ⟩ ) )
40		op1stg	⊢ ( ( ( 1 ‘ 𝑋 ) ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) ∧ 𝐹 ∈ ( 𝑌 𝐽 𝑍 ) ) → ( 1^st ‘ ⟨ ( 1 ‘ 𝑋 ) , 𝐹 ⟩ ) = ( 1 ‘ 𝑋 ) )
41	35 12 40	syl2anc	⊢ ( 𝜑 → ( 1^st ‘ ⟨ ( 1 ‘ 𝑋 ) , 𝐹 ⟩ ) = ( 1 ‘ 𝑋 ) )
42	33 39 41	3eqtrd	⊢ ( 𝜑 → ( ( 1 ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑌 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑋 , 𝑍 ⟩ ) 𝐹 ) = ( 1 ‘ 𝑋 ) )
43	19 25 42	3eqtrd	⊢ ( 𝜑 → ( ( 𝑌 ( 2^nd ‘ 𝐾 ) 𝑍 ) ‘ 𝐹 ) = ( 1 ‘ 𝑋 ) )

Metamath Proof Explorer

Theorem diag12

Proof