Theorem rnhfstructsshf 46294

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 4102	. . . . 5 ⊢ (𝐹 ∈ (dom Struct ∩ 𝒫 (V × HF )) → 𝐹 ∈ 𝒫 (V × HF ))
2	1	elpwid 4569	. . . 4 ⊢ (𝐹 ∈ (dom Struct ∩ 𝒫 (V × HF )) → 𝐹 ⊆ (V × HF ))
3		df-hfstruct 46291	. . . 4 ⊢ HFStruct = (dom Struct ∩ 𝒫 (V × HF ))
4	2, 3	eleq2s 2882	. . 3 ⊢ (𝐹 ∈ HFStruct → 𝐹 ⊆ (V × HF ))
5		rnss 5933	. . 3 ⊢ (𝐹 ⊆ (V × HF ) → ran 𝐹 ⊆ ran (V × HF ))
6	4, 5	syl 18	. 2 ⊢ (𝐹 ∈ HFStruct → ran 𝐹 ⊆ ran (V × HF ))
7		rnxpss 6182	. 2 ⊢ ran (V × HF ) ⊆ HF
8	6, 7	sstrdi 3925	1 ⊢ (𝐹 ∈ HFStruct → ran 𝐹 ⊆ HF )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560   × cxp 5656  dom cdm 5658  ran crn 5659   HF chf 9941   Struct cstr 17368  HFStructchfstruct 46290

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669  df-hfstruct 46291

This theorem is used by:  rnhfstructhf  46296