Theorem ressco

Description: comp is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017)

	Ref	Expression
Hypotheses	resshom.1	⊢ 𝐷 = ( 𝐶 ↾s 𝐴 )
	ressco.2	⊢ · = ( comp ‘ 𝐶 )
Assertion	ressco	⊢ ( 𝐴 ∈ 𝑉 → · = ( comp ‘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		resshom.1	⊢ 𝐷 = ( 𝐶 ↾s 𝐴 )
2		ressco.2	⊢ · = ( comp ‘ 𝐶 )
3		ccoid	⊢ comp = Slot ( comp ‘ ndx )
4		slotsbhcdif	⊢ ( ( Base ‘ ndx ) ≠ ( Hom ‘ ndx ) ∧ ( Base ‘ ndx ) ≠ ( comp ‘ ndx ) ∧ ( Hom ‘ ndx ) ≠ ( comp ‘ ndx ) )
5		simp2	⊢ ( ( ( Base ‘ ndx ) ≠ ( Hom ‘ ndx ) ∧ ( Base ‘ ndx ) ≠ ( comp ‘ ndx ) ∧ ( Hom ‘ ndx ) ≠ ( comp ‘ ndx ) ) → ( Base ‘ ndx ) ≠ ( comp ‘ ndx ) )
6	5	necomd	⊢ ( ( ( Base ‘ ndx ) ≠ ( Hom ‘ ndx ) ∧ ( Base ‘ ndx ) ≠ ( comp ‘ ndx ) ∧ ( Hom ‘ ndx ) ≠ ( comp ‘ ndx ) ) → ( comp ‘ ndx ) ≠ ( Base ‘ ndx ) )
7	4 6	ax-mp	⊢ ( comp ‘ ndx ) ≠ ( Base ‘ ndx )
8	1 2 3 7	resseqnbas	⊢ ( 𝐴 ∈ 𝑉 → · = ( comp ‘ 𝐷 ) )