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 ) ⟩ } )