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Metamath Proof Explorer

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 ) )