Theorem ressvalsets 12685

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 2771	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2	1	adantr 276	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑊 ∈ V)
3		elex 2771	. . 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 12677	. . . 4 ⊢ (Base‘ndx) ∈ ℕ
7	6	a1i 9	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (Base‘ndx) ∈ ℕ)
8		inex1g 4166	. . . 4 ⊢ (𝐴 ∈ 𝑌 → (𝐴 ∩ (Base‘𝑊)) ∈ V)
9	8	adantl 277	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝐴 ∩ (Base‘𝑊)) ∈ V)
10		setsex 12653	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ (Base‘ndx) ∈ ℕ ∧ (𝐴 ∩ (Base‘𝑊)) ∈ V) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) ∈ V)
11	5, 7, 9, 10	syl3anc 1249	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) ∈ V)
12		id 19	. . . 4 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
13		fveq2 5555	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
14	13	ineq2d 3361	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∩ (Base‘𝑤)) = (𝑥 ∩ (Base‘𝑊)))
15	14	opeq2d 3812	. . . 4 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩)
16	12, 15	oveq12d 5937	. . 3 ⊢ (𝑤 = 𝑊 → (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩))
17		ineq1 3354	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ (Base‘𝑊)) = (𝐴 ∩ (Base‘𝑊)))
18	17	opeq2d 3812	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩ = ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩)
19	18	oveq2d 5935	. . 3 ⊢ (𝑥 = 𝐴 → (𝑊 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
20		df-iress 12629	. . 3 ⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
21	16, 19, 20	ovmpog 6054	. 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 1249	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∩ cin 3153  ⟨cop 3622  ‘cfv 5255  (class class class)co 5919  ℕcn 8984  ndxcnx 12618   sSet csts 12619  Basecbs 12621   ↾_s cress 12622

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629

This theorem is referenced by:  ressex  12686  ressval2  12687  ressbasd  12688  strressid  12692  ressval3d  12693  resseqnbasd  12694  ressinbasd  12695  ressressg  12696  mgpress  13430