Theorem wrd3tpop

Description: A word of length three represented as triple of ordered pairs. (Contributed by AV, 26-Jan-2021)

	Ref	Expression
Assertion	wrd3tpop	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → 𝑊 = { ⟨ 0 , ( 𝑊 ‘ 0 ) ⟩ , ⟨ 1 , ( 𝑊 ‘ 1 ) ⟩ , ⟨ 2 , ( 𝑊 ‘ 2 ) ⟩ } )

Proof

Step	Hyp	Ref	Expression
1		wrdfn	⊢ ( 𝑊 ∈ Word 𝑉 → 𝑊 Fn ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
2	1	adantr	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → 𝑊 Fn ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
3		oveq2	⊢ ( ( ♯ ‘ 𝑊 ) = 3 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = ( 0 ..^ 3 ) )
4		fzo0to3tp	⊢ ( 0 ..^ 3 ) = { 0 , 1 , 2 }
5	3 4	eqtr2di	⊢ ( ( ♯ ‘ 𝑊 ) = 3 → { 0 , 1 , 2 } = ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
6	5	adantl	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → { 0 , 1 , 2 } = ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
7	6	fneq2d	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → ( 𝑊 Fn { 0 , 1 , 2 } ↔ 𝑊 Fn ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) )
8	2 7	mpbird	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → 𝑊 Fn { 0 , 1 , 2 } )
9		c0ex	⊢ 0 ∈ V
10		1ex	⊢ 1 ∈ V
11		2ex	⊢ 2 ∈ V
12	9 10 11	fntpb	⊢ ( 𝑊 Fn { 0 , 1 , 2 } ↔ 𝑊 = { ⟨ 0 , ( 𝑊 ‘ 0 ) ⟩ , ⟨ 1 , ( 𝑊 ‘ 1 ) ⟩ , ⟨ 2 , ( 𝑊 ‘ 2 ) ⟩ } )
13	8 12	sylib	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → 𝑊 = { ⟨ 0 , ( 𝑊 ‘ 0 ) ⟩ , ⟨ 1 , ( 𝑊 ‘ 1 ) ⟩ , ⟨ 2 , ( 𝑊 ‘ 2 ) ⟩ } )