Theorem ressvalsets 13146

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 2814	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2	1	adantr 276	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑊 ∈ V)
3		elex 2814	. . 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 13137	. . . 4 ⊢ (Base‘ndx) ∈ ℕ
7	6	a1i 9	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (Base‘ndx) ∈ ℕ)
8		inex1g 4225	. . . 4 ⊢ (𝐴 ∈ 𝑌 → (𝐴 ∩ (Base‘𝑊)) ∈ V)
9	8	adantl 277	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝐴 ∩ (Base‘𝑊)) ∈ V)
10		setsex 13113	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ (Base‘ndx) ∈ ℕ ∧ (𝐴 ∩ (Base‘𝑊)) ∈ V) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) ∈ V)
11	5, 7, 9, 10	syl3anc 1273	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) ∈ V)
12		id 19	. . . 4 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
13		fveq2 5639	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
14	13	ineq2d 3408	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∩ (Base‘𝑤)) = (𝑥 ∩ (Base‘𝑊)))
15	14	opeq2d 3869	. . . 4 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩)
16	12, 15	oveq12d 6035	. . 3 ⊢ (𝑤 = 𝑊 → (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩))
17		ineq1 3401	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ (Base‘𝑊)) = (𝐴 ∩ (Base‘𝑊)))
18	17	opeq2d 3869	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩ = ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩)
19	18	oveq2d 6033	. . 3 ⊢ (𝑥 = 𝐴 → (𝑊 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
20		df-iress 13089	. . 3 ⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
21	16, 19, 20	ovmpog 6155	. 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 1273	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ∩ cin 3199  ⟨cop 3672  ‘cfv 5326  (class class class)co 6017  ℕcn 9142  ndxcnx 13078   sSet csts 13079  Basecbs 13081   ↾_s cress 13082

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089

This theorem is referenced by:  ressex  13147  ressval2  13148  ressbasd  13149  strressid  13153  ressval3d  13154  resseqnbasd  13155  ressinbasd  13156  ressressg  13157  mgpress  13943