Theorem ishfstruct 46164

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 17335	. . . . 5 ⊢ Rel Struct
2	1	releldmi 5931	. . . 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 17337	. . . . 5 ⊢ (𝐹 Struct 𝑋 → 𝐹 ∈ V)
5	4	3ad2ant1 1151	. . . 4 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ V)
6		relssdmrn 6274	. . . . . 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 3933	. . . . . . 7 ⊢ dom 𝐹 ⊆ V
9		xpss12 5666	. . . . . . 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 3919	. . . 4 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ⊆ (V × HF ))
13	5, 12	elpwd 4562	. . 3 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ 𝒫 (V × HF ))
14	3, 13	elind 4096	. 2 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ (dom Struct ∩ 𝒫 (V × HF )))
15		df-hfstruct 46160	. 2 ⊢ HFStruct = (dom Struct ∩ 𝒫 (V × HF ))
16	14, 15	eleqtrrdi 2871	1 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ HFStruct)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088  𝒫 cpw 4556   class class class wbr 5102   × cxp 5649  dom cdm 5651  ran crn 5652  Rel wrel 5656   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  ax-sep 5248  ax-pr 5391

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-ral 3077  df-rex 3087  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-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-struct 17334  df-hfstruct 46160

This theorem is used by: (None)