Theorem wrdf1d

Description: A one-to-one word maps its domain into its alphabet. (Contributed by Mingli Yuan, 11-Aug-2026)

	Ref	Expression
Hypotheses	wrdf1d.w	⊢ ( 𝜑 → 𝑊 ∈ Word 𝐷 )
	wrdf1d.f	⊢ ( 𝜑 → Fun ◡ 𝑊 )
Assertion	wrdf1d	⊢ ( 𝜑 → 𝑊 : dom 𝑊 –1-1→ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		wrdf1d.w	⊢ ( 𝜑 → 𝑊 ∈ Word 𝐷 )
2		wrdf1d.f	⊢ ( 𝜑 → Fun ◡ 𝑊 )
3		wrddm	⊢ ( 𝑊 ∈ Word 𝐷 → dom 𝑊 = ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
4	1 3	syl	⊢ ( 𝜑 → dom 𝑊 = ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
5	4	eqcomd	⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = dom 𝑊 )
6		wrdf	⊢ ( 𝑊 ∈ Word 𝐷 → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝐷 )
7	1 6	syl	⊢ ( 𝜑 → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝐷 )
8	5 7	feq2dd	⊢ ( 𝜑 → 𝑊 : dom 𝑊 ⟶ 𝐷 )
9		df-f1	⊢ ( 𝑊 : dom 𝑊 –1-1→ 𝐷 ↔ ( 𝑊 : dom 𝑊 ⟶ 𝐷 ∧ Fun ◡ 𝑊 ) )
10	9	a1i	⊢ ( 𝜑 → ( 𝑊 : dom 𝑊 –1-1→ 𝐷 ↔ ( 𝑊 : dom 𝑊 ⟶ 𝐷 ∧ Fun ◡ 𝑊 ) ) )
11	8 2 10	mpbir2and	⊢ ( 𝜑 → 𝑊 : dom 𝑊 –1-1→ 𝐷 )