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Metamath Proof Explorer

Theorem wrd2pr2op

Description: A word of length two represented as unordered pair of ordered pairs. (Contributed by AV, 20-Oct-2018) (Proof shortened by AV, 26-Jan-2021)

Ref Expression
Assertion wrd2pr2op ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 2 ) → 𝑊 = { ⟨ 0 , ( 𝑊 ‘ 0 ) ⟩ , ⟨ 1 , ( 𝑊 ‘ 1 ) ⟩ } )

Proof

Step Hyp Ref Expression
1 wrdfn ⊢ ( 𝑊 ∈ Word 𝑉 → 𝑊 Fn ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
2 1 adantr ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 2 ) → 𝑊 Fn ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
3 oveq2 ⊢ ( ( ♯ ‘ 𝑊 ) = 2 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = ( 0 ..^ 2 ) )
4 fzo0to2pr ⊢ ( 0 ..^ 2 ) = { 0 , 1 }
5 3 4 eqtr2di ⊢ ( ( ♯ ‘ 𝑊 ) = 2 → { 0 , 1 } = ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
6 5 adantl ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 2 ) → { 0 , 1 } = ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
7 6 fneq2d ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 2 ) → ( 𝑊 Fn { 0 , 1 } ↔ 𝑊 Fn ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) )
8 2 7 mpbird ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 2 ) → 𝑊 Fn { 0 , 1 } )
9 c0ex ⊢ 0 ∈ V
10 1ex ⊢ 1 ∈ V
11 9 10 fnprb ⊢ ( 𝑊 Fn { 0 , 1 } ↔ 𝑊 = { ⟨ 0 , ( 𝑊 ‘ 0 ) ⟩ , ⟨ 1 , ( 𝑊 ‘ 1 ) ⟩ } )
12 8 11 sylib ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 2 ) → 𝑊 = { ⟨ 0 , ( 𝑊 ‘ 0 ) ⟩ , ⟨ 1 , ( 𝑊 ‘ 1 ) ⟩ } )