diag2

Description: Value of the diagonal functor at a morphism. (Contributed by Mario Carneiro, 7-Jan-2017)

	Ref	Expression
Hypotheses	diag2.l	⊢ 𝐿 = ( 𝐶 Δ_func 𝐷 )
	diag2.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
	diag2.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
	diag2.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	diag2.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	diag2.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
	diag2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
	diag2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐴 )
	diag2.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
Assertion	diag2	⊢ ( 𝜑 → ( ( 𝑋 ( 2^nd ‘ 𝐿 ) 𝑌 ) ‘ 𝐹 ) = ( 𝐵 × { 𝐹 } ) )

Proof

Step	Hyp	Ref	Expression
1		diag2.l	⊢ 𝐿 = ( 𝐶 Δ_func 𝐷 )
2		diag2.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
3		diag2.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
4		diag2.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
5		diag2.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
6		diag2.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
7		diag2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
8		diag2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐴 )
9		diag2.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
10	1 5 6	diagval	⊢ ( 𝜑 → 𝐿 = ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) )
11	10	fveq2d	⊢ ( 𝜑 → ( 2^nd ‘ 𝐿 ) = ( 2^nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) )
12	11	oveqd	⊢ ( 𝜑 → ( 𝑋 ( 2^nd ‘ 𝐿 ) 𝑌 ) = ( 𝑋 ( 2^nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) 𝑌 ) )
13	12	fveq1d	⊢ ( 𝜑 → ( ( 𝑋 ( 2^nd ‘ 𝐿 ) 𝑌 ) ‘ 𝐹 ) = ( ( 𝑋 ( 2^nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) 𝑌 ) ‘ 𝐹 ) )
14		eqid	⊢ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) = ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) )
15		eqid	⊢ ( 𝐶 ×_c 𝐷 ) = ( 𝐶 ×_c 𝐷 )
16		eqid	⊢ ( 𝐶 1^st_F 𝐷 ) = ( 𝐶 1^st_F 𝐷 )
17	15 5 6 16	1stfcl	⊢ ( 𝜑 → ( 𝐶 1^st_F 𝐷 ) ∈ ( ( 𝐶 ×_c 𝐷 ) Func 𝐶 ) )
18		eqid	⊢ ( Id ‘ 𝐷 ) = ( Id ‘ 𝐷 )
19		eqid	⊢ ( ( 𝑋 ( 2^nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) 𝑌 ) ‘ 𝐹 ) = ( ( 𝑋 ( 2^nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) 𝑌 ) ‘ 𝐹 )
20	14 2 5 6 17 3 4 18 7 8 9 19	curf2	⊢ ( 𝜑 → ( ( 𝑋 ( 2^nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐶 1^st_F 𝐷 ) ) ) 𝑌 ) ‘ 𝐹 ) = ( 𝑥 ∈ 𝐵 ↦ ( 𝐹 ( ⟨ 𝑋 , 𝑥 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ) ) )
21	15 2 3	xpcbas	⊢ ( 𝐴 × 𝐵 ) = ( Base ‘ ( 𝐶 ×_c 𝐷 ) )
22		eqid	⊢ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) = ( Hom ‘ ( 𝐶 ×_c 𝐷 ) )
23	5	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 ∈ Cat )
24	6	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐷 ∈ Cat )
25		opelxpi	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) → ⟨ 𝑋 , 𝑥 ⟩ ∈ ( 𝐴 × 𝐵 ) )
26	7 25	sylan	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ⟨ 𝑋 , 𝑥 ⟩ ∈ ( 𝐴 × 𝐵 ) )
27		opelxpi	⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) → ⟨ 𝑌 , 𝑥 ⟩ ∈ ( 𝐴 × 𝐵 ) )
28	8 27	sylan	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ⟨ 𝑌 , 𝑥 ⟩ ∈ ( 𝐴 × 𝐵 ) )
29	15 21 22 23 24 16 26 28	1stf2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ⟨ 𝑋 , 𝑥 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) = ( 1^st ↾ ( ⟨ 𝑋 , 𝑥 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) ) )
30	29	oveqd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 ( ⟨ 𝑋 , 𝑥 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ) = ( 𝐹 ( 1^st ↾ ( ⟨ 𝑋 , 𝑥 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) ) ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ) )
31		df-ov	⊢ ( 𝐹 ( 1^st ↾ ( ⟨ 𝑋 , 𝑥 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) ) ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ) = ( ( 1^st ↾ ( ⟨ 𝑋 , 𝑥 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) ) ‘ ⟨ 𝐹 , ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ⟩ )
32	9	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
33		eqid	⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
34		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 )
35	3 33 18 24 34	catidcl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) )
36	32 35	opelxpd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ⟨ 𝐹 , ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ⟩ ∈ ( ( 𝑋 𝐻 𝑌 ) × ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ) )
37	7	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑋 ∈ 𝐴 )
38	8	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑌 ∈ 𝐴 )
39	15 2 3 4 33 37 34 38 34 22	xpchom2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ⟨ 𝑋 , 𝑥 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) = ( ( 𝑋 𝐻 𝑌 ) × ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ) )
40	36 39	eleqtrrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ⟨ 𝐹 , ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ⟩ ∈ ( ⟨ 𝑋 , 𝑥 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) )
41	40	fvresd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( 1^st ↾ ( ⟨ 𝑋 , 𝑥 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) ) ‘ ⟨ 𝐹 , ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ⟩ ) = ( 1^st ‘ ⟨ 𝐹 , ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ⟩ ) )
42	31 41	eqtrid	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 ( 1^st ↾ ( ⟨ 𝑋 , 𝑥 ⟩ ( Hom ‘ ( 𝐶 ×_c 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) ) ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ) = ( 1^st ‘ ⟨ 𝐹 , ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ⟩ ) )
43		op1stg	⊢ ( ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ) → ( 1^st ‘ ⟨ 𝐹 , ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ⟩ ) = 𝐹 )
44	9 35 43	syl2an2r	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 1^st ‘ ⟨ 𝐹 , ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ⟩ ) = 𝐹 )
45	30 42 44	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 ( ⟨ 𝑋 , 𝑥 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ) = 𝐹 )
46	45	mpteq2dva	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ ( 𝐹 ( ⟨ 𝑋 , 𝑥 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐵 ↦ 𝐹 ) )
47		fconstmpt	⊢ ( 𝐵 × { 𝐹 } ) = ( 𝑥 ∈ 𝐵 ↦ 𝐹 )
48	46 47	eqtr4di	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ ( 𝐹 ( ⟨ 𝑋 , 𝑥 ⟩ ( 2^nd ‘ ( 𝐶 1^st_F 𝐷 ) ) ⟨ 𝑌 , 𝑥 ⟩ ) ( ( Id ‘ 𝐷 ) ‘ 𝑥 ) ) ) = ( 𝐵 × { 𝐹 } ) )
49	13 20 48	3eqtrd	⊢ ( 𝜑 → ( ( 𝑋 ( 2^nd ‘ 𝐿 ) 𝑌 ) ‘ 𝐹 ) = ( 𝐵 × { 𝐹 } ) )

Metamath Proof Explorer

Theorem diag2

Proof