Theorem hfdm

Description: The domain of a hereditarily finite set is hereditarily finite. (Contributed by Eric Schmidt, 26-Sep-2026)

	Ref	Expression
Assertion	hfdm	Could not format assertion : No typesetting found for |- ( A e. HF -> dom A e. HF ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		hfuni	Could not format ( A e. HF -> U. A e. HF ) : No typesetting found for |- ( A e. HF -> U. A e. HF ) with typecode |-
2		hfuni	Could not format ( U. A e. HF -> U. U. A e. HF ) : No typesetting found for |- ( U. A e. HF -> U. U. A e. HF ) with typecode |-
3		ssun1	$$⊢\mathrm{dom}A\subseteq \mathrm{dom}A\cup \mathrm{ran}A$$
4		dmrnssfld	$$⊢\mathrm{dom}A\cup \mathrm{ran}A\subseteq \bigcup \bigcup A$$
5	3 4	sstri	$$⊢\mathrm{dom}A\subseteq \bigcup \bigcup A$$
6		hfsshf	Could not format ( ( dom A C_ U. U. A /\ U. U. A e. HF ) -> dom A e. HF ) : No typesetting found for |- ( ( dom A C_ U. U. A /\ U. U. A e. HF ) -> dom A e. HF ) with typecode |-
7	5 6	mpan	Could not format ( U. U. A e. HF -> dom A e. HF ) : No typesetting found for |- ( U. U. A e. HF -> dom A e. HF ) with typecode |-
8	1 2 7	3syl	Could not format ( A e. HF -> dom A e. HF ) : No typesetting found for |- ( A e. HF -> dom A e. HF ) with typecode |-