Theorem reprfi2

Description: Corollary of reprinfz1 . (Contributed by Thierry Arnoux, 15-Dec-2021)

	Ref	Expression
Hypotheses	reprinfz1.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
	reprinfz1.s	⊢ ( 𝜑 → 𝑆 ∈ ℕ0 )
	reprinfz1.a	⊢ ( 𝜑 → 𝐴 ⊆ ℕ )
Assertion	reprfi2	⊢ ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑁 ) ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		reprinfz1.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
2		reprinfz1.s	⊢ ( 𝜑 → 𝑆 ∈ ℕ0 )
3		reprinfz1.a	⊢ ( 𝜑 → 𝐴 ⊆ ℕ )
4	1 2 3	reprinfz1	⊢ ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑁 ) = ( ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( repr ‘ 𝑆 ) 𝑁 ) )
5		inss2	⊢ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ⊆ ( 1 ... 𝑁 )
6		fz1ssnn	⊢ ( 1 ... 𝑁 ) ⊆ ℕ
7	5 6	sstri	⊢ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ⊆ ℕ
8	7	a1i	⊢ ( 𝜑 → ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ⊆ ℕ )
9	1	nn0zd	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
10		fzfi	⊢ ( 1 ... 𝑁 ) ∈ Fin
11	10	a1i	⊢ ( 𝜑 → ( 1 ... 𝑁 ) ∈ Fin )
12	5	a1i	⊢ ( 𝜑 → ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ⊆ ( 1 ... 𝑁 ) )
13	11 12	ssfid	⊢ ( 𝜑 → ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ∈ Fin )
14	8 9 2 13	reprfi	⊢ ( 𝜑 → ( ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( repr ‘ 𝑆 ) 𝑁 ) ∈ Fin )
15	4 14	eqeltrd	⊢ ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑁 ) ∈ Fin )