Theorem wrddin

Description: A word in two alphabets is also a word under their intersection. (Contributed by Ender Ting, 24-Jul-2026)

	Ref	Expression
Assertion	wrddin	⊢ ( ( 𝐴 ∈ Word 𝐵 ∧ 𝐴 ∈ Word 𝐶 ) → 𝐴 ∈ Word ( 𝐵 ∩ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		wrdf	⊢ ( 𝐴 ∈ Word 𝐵 → 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ 𝐵 )
2		wrdf	⊢ ( 𝐴 ∈ Word 𝐶 → 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ 𝐶 )
3	1 2	anim12i	⊢ ( ( 𝐴 ∈ Word 𝐵 ∧ 𝐴 ∈ Word 𝐶 ) → ( 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ 𝐵 ∧ 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ 𝐶 ) )
4		fin	⊢ ( 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ 𝐵 ∧ 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ 𝐶 ) )
5	3 4	sylibr	⊢ ( ( 𝐴 ∈ Word 𝐵 ∧ 𝐴 ∈ Word 𝐶 ) → 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ ( 𝐵 ∩ 𝐶 ) )
6		iswrdb	⊢ ( 𝐴 ∈ Word ( 𝐵 ∩ 𝐶 ) ↔ 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ ( 𝐵 ∩ 𝐶 ) )
7	5 6	sylibr	⊢ ( ( 𝐴 ∈ Word 𝐵 ∧ 𝐴 ∈ Word 𝐶 ) → 𝐴 ∈ Word ( 𝐵 ∩ 𝐶 ) )