Theorem ressval3d 13426

Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)

Hypotheses

Ref	Expression
ressval3d.r	⊢ 𝑅 = (𝑆 ↾s 𝐴)
ressval3d.b	⊢ 𝐵 = (Base‘𝑆)
ressval3d.e	⊢ 𝐸 = (Base‘ndx)
ressval3d.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
ressval3d.f	⊢ (𝜑 → Fun 𝑆)
ressval3d.d	⊢ (𝜑 → 𝐸 ∈ dom 𝑆)
ressval3d.u	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
ressval3d	⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Proof of Theorem ressval3d

Step	Hyp	Ref	Expression
1		ressval3d.r	. . 3 ⊢ 𝑅 = (𝑆 ↾s 𝐴)
2		ressval3d.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
3		ressval3d.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
4		basfn 13411	. . . . . . 7 ⊢ Base Fn V
5	2	elexd 2835	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V)
6		funfvex 5712	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
7	6	funfni 5483	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
8	4, 5, 7	sylancr 418	. . . . . 6 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
9	3, 8	eqeltrid 2325	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
10		ressval3d.u	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
11	9, 10	ssexd 4273	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
12		ressvalsets 13418	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝑆 ↾s 𝐴) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
13	2, 11, 12	syl2anc 415	. . 3 ⊢ (𝜑 → (𝑆 ↾s 𝐴) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
14	1, 13	eqtrid 2283	. 2 ⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
15		ressval3d.e	. . . . 5 ⊢ 𝐸 = (Base‘ndx)
16	15	a1i 9	. . . 4 ⊢ (𝜑 → 𝐸 = (Base‘ndx))
17		df-ss 3233	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
18	10, 17	sylib 122	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐴)
19	3	ineq2i 3429	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ (Base‘𝑆))
20	18, 19	eqtr3di 2286	. . . 4 ⊢ (𝜑 → 𝐴 = (𝐴 ∩ (Base‘𝑆)))
21	16, 20	opeq12d 3912	. . 3 ⊢ (𝜑 → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩)
22	21	oveq2d 6101	. 2 ⊢ (𝜑 → (𝑆 sSet ⟨𝐸, 𝐴⟩) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
23	14, 22	eqtr4d 2274	1 ⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220  ⟨cop 3712  dom cdm 4774  Fun wfun 5371   Fn wfn 5372  ‘cfv 5377  (class class class)co 6085  ndxcnx 13349   sSet csts 13350  Basecbs 13352   ↾_s cress 13353

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-inn 9305  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360

This theorem is used by: (None)