Theorem ressvalsets 13137

Description: Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)

Assertion

Ref	Expression
ressvalsets	⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))

Proof of Theorem ressvalsets

Dummy variables 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2812	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2	1	adantr 276	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑊 ∈ V)
3		elex 2812	. . 3 ⊢ (𝐴 ∈ 𝑌 → 𝐴 ∈ V)
4	3	adantl 277	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝐴 ∈ V)
5		simpl 109	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑊 ∈ 𝑋)
6		basendxnn 13128	. . . 4 ⊢ (Base‘ndx) ∈ ℕ
7	6	a1i 9	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (Base‘ndx) ∈ ℕ)
8		inex1g 4223	. . . 4 ⊢ (𝐴 ∈ 𝑌 → (𝐴 ∩ (Base‘𝑊)) ∈ V)
9	8	adantl 277	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝐴 ∩ (Base‘𝑊)) ∈ V)
10		setsex 13104	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ (Base‘ndx) ∈ ℕ ∧ (𝐴 ∩ (Base‘𝑊)) ∈ V) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) ∈ V)
11	5, 7, 9, 10	syl3anc 1271	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) ∈ V)
12		id 19	. . . 4 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
13		fveq2 5635	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
14	13	ineq2d 3406	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∩ (Base‘𝑤)) = (𝑥 ∩ (Base‘𝑊)))
15	14	opeq2d 3867	. . . 4 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩)
16	12, 15	oveq12d 6031	. . 3 ⊢ (𝑤 = 𝑊 → (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩))
17		ineq1 3399	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ (Base‘𝑊)) = (𝐴 ∩ (Base‘𝑊)))
18	17	opeq2d 3867	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩ = ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩)
19	18	oveq2d 6029	. . 3 ⊢ (𝑥 = 𝐴 → (𝑊 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
20		df-iress 13080	. . 3 ⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
21	16, 19, 20	ovmpog 6151	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V ∧ (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
22	2, 4, 11, 21	syl3anc 1271	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2800   ∩ cin 3197  ⟨cop 3670  ‘cfv 5324  (class class class)co 6013  ℕcn 9133  ndxcnx 13069   sSet csts 13070  Basecbs 13072   ↾_s cress 13073

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-inn 9134  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080

This theorem is referenced by:  ressex  13138  ressval2  13139  ressbasd  13140  strressid  13144  ressval3d  13145  resseqnbasd  13146  ressinbasd  13147  ressressg  13148  mgpress  13934