Theorem hfstructstruct 46024

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 4147	. . 3 ⊢ (𝐹 ∈ (dom Struct ∩ 𝒫 (V × HF )) → 𝐹 ∈ dom Struct )
2		df-hfstruct 46023	. . 3 ⊢ HFStruct = (dom Struct ∩ 𝒫 (V × HF ))
3	1, 2	eleq2s 2879	. 2 ⊢ (𝐹 ∈ HFStruct → 𝐹 ∈ dom Struct )
4		eldmg 5880	. . 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 2145  Vcvv 3451   ∩ cin 3898  𝒫 cpw 4557   class class class wbr 5103   × cxp 5649  dom cdm 5651   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

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-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-dm 5661  df-hfstruct 46023

This theorem is used by:  hfstructfun  46025  rnhfstructhf  46028  hfstructhf  46029  hfstructcan  46030