Theorem fzprval

Description: Two ways of defining the first two values of a sequence on NN . (Contributed by NM, 5-Sep-2011)

	Ref	Expression
Assertion	fzprval	⊢ ( ∀ 𝑥 ∈ ( 1 ... 2 ) ( 𝐹 ‘ 𝑥 ) = if ( 𝑥 = 1 , 𝐴 , 𝐵 ) ↔ ( ( 𝐹 ‘ 1 ) = 𝐴 ∧ ( 𝐹 ‘ 2 ) = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fz12pr	⊢ ( 1 ... 2 ) = { 1 , 2 }
2	1	raleqi	⊢ ( ∀ 𝑥 ∈ ( 1 ... 2 ) ( 𝐹 ‘ 𝑥 ) = if ( 𝑥 = 1 , 𝐴 , 𝐵 ) ↔ ∀ 𝑥 ∈ { 1 , 2 } ( 𝐹 ‘ 𝑥 ) = if ( 𝑥 = 1 , 𝐴 , 𝐵 ) )
3		1ex	⊢ 1 ∈ V
4		2ex	⊢ 2 ∈ V
5		fveq2	⊢ ( 𝑥 = 1 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 1 ) )
6		iftrue	⊢ ( 𝑥 = 1 → if ( 𝑥 = 1 , 𝐴 , 𝐵 ) = 𝐴 )
7	5 6	eqeq12d	⊢ ( 𝑥 = 1 → ( ( 𝐹 ‘ 𝑥 ) = if ( 𝑥 = 1 , 𝐴 , 𝐵 ) ↔ ( 𝐹 ‘ 1 ) = 𝐴 ) )
8		fveq2	⊢ ( 𝑥 = 2 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 2 ) )
9		1ne2	⊢ 1 ≠ 2
10	9	necomi	⊢ 2 ≠ 1
11		pm13.181	⊢ ( ( 𝑥 = 2 ∧ 2 ≠ 1 ) → 𝑥 ≠ 1 )
12	10 11	mpan2	⊢ ( 𝑥 = 2 → 𝑥 ≠ 1 )
13	12	neneqd	⊢ ( 𝑥 = 2 → ¬ 𝑥 = 1 )
14	13	iffalsed	⊢ ( 𝑥 = 2 → if ( 𝑥 = 1 , 𝐴 , 𝐵 ) = 𝐵 )
15	8 14	eqeq12d	⊢ ( 𝑥 = 2 → ( ( 𝐹 ‘ 𝑥 ) = if ( 𝑥 = 1 , 𝐴 , 𝐵 ) ↔ ( 𝐹 ‘ 2 ) = 𝐵 ) )
16	3 4 7 15	ralpr	⊢ ( ∀ 𝑥 ∈ { 1 , 2 } ( 𝐹 ‘ 𝑥 ) = if ( 𝑥 = 1 , 𝐴 , 𝐵 ) ↔ ( ( 𝐹 ‘ 1 ) = 𝐴 ∧ ( 𝐹 ‘ 2 ) = 𝐵 ) )
17	2 16	bitri	⊢ ( ∀ 𝑥 ∈ ( 1 ... 2 ) ( 𝐹 ‘ 𝑥 ) = if ( 𝑥 = 1 , 𝐴 , 𝐵 ) ↔ ( ( 𝐹 ‘ 1 ) = 𝐴 ∧ ( 𝐹 ‘ 2 ) = 𝐵 ) )