Theorem ressvalsets 12537

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 2760	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2	1	adantr 276	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑊 ∈ V)
3		elex 2760	. . 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 12531	. . . 4 ⊢ (Base‘ndx) ∈ ℕ
7	6	a1i 9	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (Base‘ndx) ∈ ℕ)
8		inex1g 4151	. . . 4 ⊢ (𝐴 ∈ 𝑌 → (𝐴 ∩ (Base‘𝑊)) ∈ V)
9	8	adantl 277	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝐴 ∩ (Base‘𝑊)) ∈ V)
10		setsex 12507	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ (Base‘ndx) ∈ ℕ ∧ (𝐴 ∩ (Base‘𝑊)) ∈ V) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) ∈ V)
11	5, 7, 9, 10	syl3anc 1248	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) ∈ V)
12		id 19	. . . 4 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
13		fveq2 5527	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
14	13	ineq2d 3348	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∩ (Base‘𝑤)) = (𝑥 ∩ (Base‘𝑊)))
15	14	opeq2d 3797	. . . 4 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩)
16	12, 15	oveq12d 5906	. . 3 ⊢ (𝑤 = 𝑊 → (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩))
17		ineq1 3341	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ (Base‘𝑊)) = (𝐴 ∩ (Base‘𝑊)))
18	17	opeq2d 3797	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩ = ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩)
19	18	oveq2d 5904	. . 3 ⊢ (𝑥 = 𝐴 → (𝑊 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
20		df-iress 12483	. . 3 ⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
21	16, 19, 20	ovmpog 6022	. 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 1248	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1363   ∈ wcel 2158  Vcvv 2749   ∩ cin 3140  ⟨cop 3607  ‘cfv 5228  (class class class)co 5888  ℕcn 8932  ndxcnx 12472   sSet csts 12473  Basecbs 12475   ↾_s cress 12476

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-inn 8933  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-iress 12483

This theorem is referenced by:  ressex  12538  ressval2  12539  ressbasd  12540  strressid  12544  ressval3d  12545  resseqnbasd  12546  ressinbasd  12547  ressressg  12548  mgpress  13173