Theorem ishfstruct 46027

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 17326	. . . . 5 ⊢ Rel Struct
2	1	releldmi 5930	. . . 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 17328	. . . . 5 ⊢ (𝐹 Struct 𝑋 → 𝐹 ∈ V)
5	4	3ad2ant1 1151	. . . 4 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ V)
6		relssdmrn 6271	. . . . . 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 3955	. . . . . . 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 3941	. . . 4 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ⊆ (V × HF ))
13	5, 12	elpwd 4563	. . 3 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ 𝒫 (V × HF ))
14	3, 13	elind 4146	. 2 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ (dom Struct ∩ 𝒫 (V × HF )))
15		df-hfstruct 46023	. 2 ⊢ HFStruct = (dom Struct ∩ 𝒫 (V × HF ))
16	14, 15	eleqtrrdi 2872	1 ⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ HFStruct)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557   class class class wbr 5103   × cxp 5649  dom cdm 5651  ran crn 5652  Rel wrel 5656   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  ax-sep 5249  ax-pr 5391

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-ral 3078  df-rex 3088  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-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-struct 17325  df-hfstruct 46023

This theorem is used by: (None)