Theorem fargshiftfv

Description: If a class is a function, then the values of the "shifted function" correspond to the function values of the class. (Contributed by Alexander van der Vekens, 23-Nov-2017)

	Ref	Expression
Hypothesis	fargshift.g	⊢ 𝐺 = ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↦ ( 𝐹 ‘ ( 𝑥 + 1 ) ) )
Assertion	fargshiftfv	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) → ( 𝑋 ∈ ( 0 ..^ 𝑁 ) → ( 𝐺 ‘ 𝑋 ) = ( 𝐹 ‘ ( 𝑋 + 1 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		fargshift.g	⊢ 𝐺 = ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↦ ( 𝐹 ‘ ( 𝑥 + 1 ) ) )
2		ffn	⊢ ( 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 → 𝐹 Fn ( 1 ... 𝑁 ) )
3		fseq1hash	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 Fn ( 1 ... 𝑁 ) ) → ( ♯ ‘ 𝐹 ) = 𝑁 )
4		oveq2	⊢ ( 𝑁 = ( ♯ ‘ 𝐹 ) → ( 0 ..^ 𝑁 ) = ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
5	4	eqcoms	⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( 0 ..^ 𝑁 ) = ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
6	5	eleq2d	⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( 𝑋 ∈ ( 0 ..^ 𝑁 ) ↔ 𝑋 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) )
7	6	biimpd	⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( 𝑋 ∈ ( 0 ..^ 𝑁 ) → 𝑋 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) )
8	3 7	syl	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 Fn ( 1 ... 𝑁 ) ) → ( 𝑋 ∈ ( 0 ..^ 𝑁 ) → 𝑋 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) )
9	2 8	sylan2	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) → ( 𝑋 ∈ ( 0 ..^ 𝑁 ) → 𝑋 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) )
10	9	imp	⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) ∧ 𝑋 ∈ ( 0 ..^ 𝑁 ) ) → 𝑋 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
11		fvex	⊢ ( 𝐹 ‘ ( 𝑋 + 1 ) ) ∈ V
12		fvoveq1	⊢ ( 𝑥 = 𝑋 → ( 𝐹 ‘ ( 𝑥 + 1 ) ) = ( 𝐹 ‘ ( 𝑋 + 1 ) ) )
13	12 1	fvmptg	⊢ ( ( 𝑋 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ ( 𝐹 ‘ ( 𝑋 + 1 ) ) ∈ V ) → ( 𝐺 ‘ 𝑋 ) = ( 𝐹 ‘ ( 𝑋 + 1 ) ) )
14	10 11 13	sylancl	⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) ∧ 𝑋 ∈ ( 0 ..^ 𝑁 ) ) → ( 𝐺 ‘ 𝑋 ) = ( 𝐹 ‘ ( 𝑋 + 1 ) ) )
15	14	ex	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) → ( 𝑋 ∈ ( 0 ..^ 𝑁 ) → ( 𝐺 ‘ 𝑋 ) = ( 𝐹 ‘ ( 𝑋 + 1 ) ) ) )