Theorem fzto1stfv1

Description: Value of our permutation P at 1. (Contributed by Thierry Arnoux, 23-Aug-2020)

	Ref	Expression
Hypotheses	psgnfzto1st.d	⊢ 𝐷 = ( 1 ... 𝑁 )
	psgnfzto1st.p	⊢ 𝑃 = ( 𝑖 ∈ 𝐷 ↦ if ( 𝑖 = 1 , 𝐼 , if ( 𝑖 ≤ 𝐼 , ( 𝑖 − 1 ) , 𝑖 ) ) )
Assertion	fzto1stfv1	⊢ ( 𝐼 ∈ 𝐷 → ( 𝑃 ‘ 1 ) = 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		psgnfzto1st.d	⊢ 𝐷 = ( 1 ... 𝑁 )
2		psgnfzto1st.p	⊢ 𝑃 = ( 𝑖 ∈ 𝐷 ↦ if ( 𝑖 = 1 , 𝐼 , if ( 𝑖 ≤ 𝐼 , ( 𝑖 − 1 ) , 𝑖 ) ) )
3		iftrue	⊢ ( 𝑖 = 1 → if ( 𝑖 = 1 , 𝐼 , if ( 𝑖 ≤ 𝐼 , ( 𝑖 − 1 ) , 𝑖 ) ) = 𝐼 )
4		elfzuz2	⊢ ( 𝐼 ∈ ( 1 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 1 ) )
5	4 1	eleq2s	⊢ ( 𝐼 ∈ 𝐷 → 𝑁 ∈ ( ℤ≥ ‘ 1 ) )
6		eluzfz1	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... 𝑁 ) )
7	6 1	eleqtrrdi	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ 𝐷 )
8	5 7	syl	⊢ ( 𝐼 ∈ 𝐷 → 1 ∈ 𝐷 )
9		id	⊢ ( 𝐼 ∈ 𝐷 → 𝐼 ∈ 𝐷 )
10	2 3 8 9	fvmptd3	⊢ ( 𝐼 ∈ 𝐷 → ( 𝑃 ‘ 1 ) = 𝐼 )