Theorem hfstructstruct 46161

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 4097	. . 3 ⊢ (𝐹 ∈ (dom Struct ∩ 𝒫 (V × HF )) → 𝐹 ∈ dom Struct )
2		df-hfstruct 46160	. . 3 ⊢ HFStruct = (dom Struct ∩ 𝒫 (V × HF ))
3	1, 2	eleq2s 2878	. 2 ⊢ (𝐹 ∈ HFStruct → 𝐹 ∈ dom Struct )
4		eldmg 5881	. . 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 1811   ∈ wcel 2144  Vcvv 3450   ∩ cin 4088  𝒫 cpw 4556   class class class wbr 5102   × cxp 5649  dom cdm 5651   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

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-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-dm 5661  df-hfstruct 46160

This theorem is used by:  hfstructfun  46162  rnhfstructhf  46165  hfstructhf  46166  hfstructcan  46167