Theorem ishfstruct 46295

Description: Conditions that imply that 𝐹 is an HFStruct. (Contributed by Eric Schmidt, 29-Sep-2026.)

Assertion

Ref	Expression
ishfstruct	⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ HFStruct)

Proof of Theorem ishfstruct

Step	Hyp	Ref	Expression
1		brstruct 17370	. . . . 5 ⊢ Rel Struct
2	1	releldmi 5942	. . . 4 ⊢ (𝐹 Struct 𝑋 → 𝐹 ∈ dom Struct )
3	2	3ad2ant1 1151	. . 3 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ dom Struct )
4		structex 17372	. . . . 5 ⊢ (𝐹 Struct 𝑋 → 𝐹 ∈ V)
5	4	3ad2ant1 1151	. . . 4 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ V)
6		relssdmrn 6295	. . . . . 6 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
7	6	3ad2ant2 1152	. . . . 5 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
8		ssv 3937	. . . . . . 7 ⊢ dom 𝐹 ⊆ V
9		xpss12 5673	. . . . . . 7 ⊢ ((dom 𝐹 ⊆ V ∧ ran 𝐹 ⊆ HF ) → (dom 𝐹 × ran 𝐹) ⊆ (V × HF ))
10	8, 9	mpan 703	. . . . . 6 ⊢ (ran 𝐹 ⊆ HF → (dom 𝐹 × ran 𝐹) ⊆ (V × HF ))
11	10	3ad2ant3 1153	. . . . 5 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → (dom 𝐹 × ran 𝐹) ⊆ (V × HF ))
12	7, 11	sstrd 3923	. . . 4 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ⊆ (V × HF ))
13	5, 12	elpwd 4566	. . 3 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ 𝒫 (V × HF ))
14	3, 13	elind 4100	. 2 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ (dom Struct ∩ 𝒫 (V × HF )))
15		df-hfstruct 46291	. 2 ⊢ HFStruct = (dom Struct ∩ 𝒫 (V × HF ))
16	14, 15	eleqtrrdi 2875	1 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ HFStruct)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560   class class class wbr 5108   × cxp 5656  dom cdm 5658  ran crn 5659  Rel wrel 5663   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-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-struct 17369  df-hfstruct 46291

This theorem is used by:  grpcanonhfstruct  46299