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	$$⊢\mathrm{ran}F\subseteq A\to F^{-1}\left[A\right]=\mathrm{dom}F$$

Proof

Step	Hyp	Ref	Expression
1		dfdm4	$$⊢\mathrm{dom}F=\mathrm{ran}{F}^{-1}$$
2		df-rn	$$⊢\mathrm{ran}F=\mathrm{dom}{F}^{-1}$$
3	2	sseq1i	$$⊢\mathrm{ran}F\subseteq A\leftrightarrow \mathrm{dom}{F}^{-1}\subseteq A$$
4		dfrn7	$$⊢\mathrm{dom}{F}^{-1}\subseteq A\to \mathrm{ran}{F}^{-1}=F^{-1}\left[A\right]$$
5	3 4	sylbi	$$⊢\mathrm{ran}F\subseteq A\to \mathrm{ran}{F}^{-1}=F^{-1}\left[A\right]$$
6	1 5	eqtr2id	$$⊢\mathrm{ran}F\subseteq A\to F^{-1}\left[A\right]=\mathrm{dom}F$$