Theorem rnstructfi

Description: The range of an extensible structure is finite. (Contributed by Eric Schmidt, 29-Sep-2026)

	Ref	Expression
Assertion	rnstructfi	⊢ ( 𝐹 Struct 𝑋 → ran 𝐹 ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		imadmrn	⊢ ( ◡ ◡ 𝐹 “ dom ◡ ◡ 𝐹 ) = ran ◡ ◡ 𝐹
2		rncnvcnv	⊢ ran ◡ ◡ 𝐹 = ran 𝐹
3	1 2	eqtri	⊢ ( ◡ ◡ 𝐹 “ dom ◡ ◡ 𝐹 ) = ran 𝐹
4		structfung	⊢ ( 𝐹 Struct 𝑋 → Fun ◡ ◡ 𝐹 )
5		dmcnvcnv	⊢ dom ◡ ◡ 𝐹 = dom 𝐹
6		dmstructfi	⊢ ( 𝐹 Struct 𝑋 → dom 𝐹 ∈ Fin )
7	5 6	eqeltrid	⊢ ( 𝐹 Struct 𝑋 → dom ◡ ◡ 𝐹 ∈ Fin )
8		imafi	⊢ ( ( Fun ◡ ◡ 𝐹 ∧ dom ◡ ◡ 𝐹 ∈ Fin ) → ( ◡ ◡ 𝐹 “ dom ◡ ◡ 𝐹 ) ∈ Fin )
9	4 7 8	syl2anc	⊢ ( 𝐹 Struct 𝑋 → ( ◡ ◡ 𝐹 “ dom ◡ ◡ 𝐹 ) ∈ Fin )
10	3 9	eqeltrrid	⊢ ( 𝐹 Struct 𝑋 → ran 𝐹 ∈ Fin )