Theorem rescco

Description: Composition in the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017) (Proof shortened by AV, 13-Oct-2024)

	Ref	Expression
Hypotheses	rescbas.d	⊢ 𝐷 = ( 𝐶 ↾cat 𝐻 )
	rescbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	rescbas.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
	rescbas.h	⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) )
	rescbas.s	⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
	rescco.o	⊢ · = ( comp ‘ 𝐶 )
Assertion	rescco	⊢ ( 𝜑 → · = ( comp ‘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		rescbas.d	⊢ 𝐷 = ( 𝐶 ↾cat 𝐻 )
2		rescbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		rescbas.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
4		rescbas.h	⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) )
5		rescbas.s	⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
6		rescco.o	⊢ · = ( comp ‘ 𝐶 )
7		ccoid	⊢ comp = Slot ( comp ‘ ndx )
8		slotsbhcdif	⊢ ( ( Base ‘ ndx ) ≠ ( Hom ‘ ndx ) ∧ ( Base ‘ ndx ) ≠ ( comp ‘ ndx ) ∧ ( Hom ‘ ndx ) ≠ ( comp ‘ ndx ) )
9		simp3	⊢ ( ( ( Base ‘ ndx ) ≠ ( Hom ‘ ndx ) ∧ ( Base ‘ ndx ) ≠ ( comp ‘ ndx ) ∧ ( Hom ‘ ndx ) ≠ ( comp ‘ ndx ) ) → ( Hom ‘ ndx ) ≠ ( comp ‘ ndx ) )
10	9	necomd	⊢ ( ( ( Base ‘ ndx ) ≠ ( Hom ‘ ndx ) ∧ ( Base ‘ ndx ) ≠ ( comp ‘ ndx ) ∧ ( Hom ‘ ndx ) ≠ ( comp ‘ ndx ) ) → ( comp ‘ ndx ) ≠ ( Hom ‘ ndx ) )
11	8 10	ax-mp	⊢ ( comp ‘ ndx ) ≠ ( Hom ‘ ndx )
12	7 11	setsnid	⊢ ( comp ‘ ( 𝐶 ↾s 𝑆 ) ) = ( comp ‘ ( ( 𝐶 ↾s 𝑆 ) sSet ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ ) )
13	2	fvexi	⊢ 𝐵 ∈ V
14	13	ssex	⊢ ( 𝑆 ⊆ 𝐵 → 𝑆 ∈ V )
15	5 14	syl	⊢ ( 𝜑 → 𝑆 ∈ V )
16		eqid	⊢ ( 𝐶 ↾s 𝑆 ) = ( 𝐶 ↾s 𝑆 )
17	16 6	ressco	⊢ ( 𝑆 ∈ V → · = ( comp ‘ ( 𝐶 ↾s 𝑆 ) ) )
18	15 17	syl	⊢ ( 𝜑 → · = ( comp ‘ ( 𝐶 ↾s 𝑆 ) ) )
19	1 3 15 4	rescval2	⊢ ( 𝜑 → 𝐷 = ( ( 𝐶 ↾s 𝑆 ) sSet ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ ) )
20	19	fveq2d	⊢ ( 𝜑 → ( comp ‘ 𝐷 ) = ( comp ‘ ( ( 𝐶 ↾s 𝑆 ) sSet ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ ) ) )
21	12 18 20	3eqtr4a	⊢ ( 𝜑 → · = ( comp ‘ 𝐷 ) )