Theorem cnvimassrndm

Description: The preimage of a superset of the range of a class is equal to the domain of the class. Generalization of cnvimarndm to supersets of the range. (Contributed by AV, 18-Sep-2024) (Proof shortened by BJ, 29-Sep-2026)

	Ref	Expression
Assertion	cnvimassrndm	⊢ ( ran 𝐹 ⊆ 𝐴 → ( ◡ 𝐹 “ 𝐴 ) = dom 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		dfdm4	⊢ dom 𝐹 = ran ◡ 𝐹
2		df-rn	⊢ ran 𝐹 = dom ◡ 𝐹
3	2	sseq1i	⊢ ( ran 𝐹 ⊆ 𝐴 ↔ dom ◡ 𝐹 ⊆ 𝐴 )
4		dfrn7	⊢ ( dom ◡ 𝐹 ⊆ 𝐴 → ran ◡ 𝐹 = ( ◡ 𝐹 “ 𝐴 ) )
5	3 4	sylbi	⊢ ( ran 𝐹 ⊆ 𝐴 → ran ◡ 𝐹 = ( ◡ 𝐹 “ 𝐴 ) )
6	1 5	eqtr2id	⊢ ( ran 𝐹 ⊆ 𝐴 → ( ◡ 𝐹 “ 𝐴 ) = dom 𝐹 )