Theorem rnhfstructsshf 46163

Description: The range of an HFStruct consists of hereditarily finite sets. (Contributed by Eric Schmidt, 29-Sep-2026.)

Assertion

Ref	Expression
rnhfstructsshf	⊢ (𝐹 ∈ HFStruct → ran 𝐹 ⊆ HF )

Proof of Theorem rnhfstructsshf

Step	Hyp	Ref	Expression
1		elinel2 4098	. . . . 5 ⊢ (𝐹 ∈ (dom Struct ∩ 𝒫 (V × HF )) → 𝐹 ∈ 𝒫 (V × HF ))
2	1	elpwid 4565	. . . 4 ⊢ (𝐹 ∈ (dom Struct ∩ 𝒫 (V × HF )) → 𝐹 ⊆ (V × HF ))
3		df-hfstruct 46160	. . . 4 ⊢ HFStruct = (dom Struct ∩ 𝒫 (V × HF ))
4	2, 3	eleq2s 2878	. . 3 ⊢ (𝐹 ∈ HFStruct → 𝐹 ⊆ (V × HF ))
5		rnss 5922	. . 3 ⊢ (𝐹 ⊆ (V × HF ) → ran 𝐹 ⊆ ran (V × HF ))
6	4, 5	syl 18	. 2 ⊢ (𝐹 ∈ HFStruct → ran 𝐹 ⊆ ran (V × HF ))
7		rnxpss 6166	. 2 ⊢ ran (V × HF ) ⊆ HF
8	6, 7	sstrdi 3921	1 ⊢ (𝐹 ∈ HFStruct → ran 𝐹 ⊆ HF )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088  𝒫 cpw 4556   × cxp 5649  dom cdm 5651  ran crn 5652   HF chf 9911   Struct cstr 17333  HFStructchfstruct 46159

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-hfstruct 46160

This theorem is used by:  rnhfstructhf  46165