Theorem ressval3d 12823

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 12809	. . . . . . 7 ⊢ Base Fn V
5	2	elexd 2784	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V)
6		funfvex 5587	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
7	6	funfni 5370	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
8	4, 5, 7	sylancr 414	. . . . . 6 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
9	3, 8	eqeltrid 2291	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
10		ressval3d.u	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
11	9, 10	ssexd 4183	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
12		ressvalsets 12815	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝑆 ↾s 𝐴) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
13	2, 11, 12	syl2anc 411	. . 3 ⊢ (𝜑 → (𝑆 ↾s 𝐴) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
14	1, 13	eqtrid 2249	. 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 3178	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
18	10, 17	sylib 122	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐴)
19	3	ineq2i 3370	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ (Base‘𝑆))
20	18, 19	eqtr3di 2252	. . . 4 ⊢ (𝜑 → 𝐴 = (𝐴 ∩ (Base‘𝑆)))
21	16, 20	opeq12d 3826	. . 3 ⊢ (𝜑 → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩)
22	21	oveq2d 5950	. 2 ⊢ (𝜑 → (𝑆 sSet ⟨𝐸, 𝐴⟩) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
23	14, 22	eqtr4d 2240	1 ⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Syntax hints:   → wi 4   = wceq 1372   ∈ wcel 2175  Vcvv 2771   ∩ cin 3164   ⊆ wss 3165  ⟨cop 3635  dom cdm 4673  Fun wfun 5262   Fn wfn 5263  ‘cfv 5268  (class class class)co 5934  ndxcnx 12748   sSet csts 12749  Basecbs 12751   ↾_s cress 12752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1re 8001  ax-addrcl 8004

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-iota 5229  df-fun 5270  df-fn 5271  df-fv 5276  df-ov 5937  df-oprab 5938  df-mpo 5939  df-inn 9019  df-ndx 12754  df-slot 12755  df-base 12757  df-sets 12758  df-iress 12759

This theorem is referenced by: (None)