Theorem rnhfstructsshf 46026

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 4148	. . . . 5 ⊢ (𝐹 ∈ (dom Struct ∩ 𝒫 (V × HF )) → 𝐹 ∈ 𝒫 (V × HF ))
2	1	elpwid 4566	. . . 4 ⊢ (𝐹 ∈ (dom Struct ∩ 𝒫 (V × HF )) → 𝐹 ⊆ (V × HF ))
3		df-hfstruct 46023	. . . 4 ⊢ HFStruct = (dom Struct ∩ 𝒫 (V × HF ))
4	2, 3	eleq2s 2879	. . 3 ⊢ (𝐹 ∈ HFStruct → 𝐹 ⊆ (V × HF ))
5		rnss 5921	. . 3 ⊢ (𝐹 ⊆ (V × HF ) → ran 𝐹 ⊆ ran (V × HF ))
6	4, 5	syl 18	. 2 ⊢ (𝐹 ∈ HFStruct → ran 𝐹 ⊆ ran (V × HF ))
7		rnxpss 6164	. 2 ⊢ ran (V × HF ) ⊆ HF
8	6, 7	sstrdi 3943	1 ⊢ (𝐹 ∈ HFStruct → ran 𝐹 ⊆ HF )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557   × cxp 5649  dom cdm 5651  ran crn 5652   HF chf 9904   Struct cstr 17324  HFStructchfstruct 46022

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-hfstruct 46023

This theorem is used by:  rnhfstructhf  46028