Theorem hfstructstruct 46292

Description: An HFStruct is an extensible structure. (Contributed by Eric Schmidt, 29-Sep-2026.)

Assertion

Ref	Expression
hfstructstruct	⊢ (𝐹 ∈ HFStruct → ∃𝑥 𝐹 Struct 𝑥)

Distinct variable group:   𝑥,𝐹

Proof of Theorem hfstructstruct

Step	Hyp	Ref	Expression
1		elinel1 4101	. . 3 ⊢ (𝐹 ∈ (dom Struct ∩ 𝒫 (V × HF )) → 𝐹 ∈ dom Struct )
2		df-hfstruct 46291	. . 3 ⊢ HFStruct = (dom Struct ∩ 𝒫 (V × HF ))
3	1, 2	eleq2s 2882	. 2 ⊢ (𝐹 ∈ HFStruct → 𝐹 ∈ dom Struct )
4		eldmg 5892	. . 3 ⊢ (𝐹 ∈ dom Struct → (𝐹 ∈ dom Struct ↔ ∃𝑥 𝐹 Struct 𝑥))
5	4	ibi 270	. 2 ⊢ (𝐹 ∈ dom Struct → ∃𝑥 𝐹 Struct 𝑥)
6	3, 5	syl 18	1 ⊢ (𝐹 ∈ HFStruct → ∃𝑥 𝐹 Struct 𝑥)

This proof depends on syntax axioms:   → wi 4  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092  𝒫 cpw 4560   class class class wbr 5108   × cxp 5656  dom cdm 5658   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

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-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-dm 5668  df-hfstruct 46291

This theorem is used by:  hfstructfun  46293  rnhfstructhf  46296  hfstructhf  46297  hfstructcan  46298