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Theorem ressvalsets 12548
Description: Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
Assertion
Ref Expression
ressvalsets ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))

Proof of Theorem ressvalsets
Dummy variables 𝑀 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2763 . . 3 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
21adantr 276 . 2 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ π‘Š ∈ V)
3 elex 2763 . . 3 (𝐴 ∈ π‘Œ β†’ 𝐴 ∈ V)
43adantl 277 . 2 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝐴 ∈ V)
5 simpl 109 . . 3 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ π‘Š ∈ 𝑋)
6 basendxnn 12542 . . . 4 (Baseβ€˜ndx) ∈ β„•
76a1i 9 . . 3 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ (Baseβ€˜ndx) ∈ β„•)
8 inex1g 4154 . . . 4 (𝐴 ∈ π‘Œ β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) ∈ V)
98adantl 277 . . 3 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) ∈ V)
10 setsex 12518 . . 3 ((π‘Š ∈ 𝑋 ∧ (Baseβ€˜ndx) ∈ β„• ∧ (𝐴 ∩ (Baseβ€˜π‘Š)) ∈ V) β†’ (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩) ∈ V)
115, 7, 9, 10syl3anc 1249 . 2 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩) ∈ V)
12 id 19 . . . 4 (𝑀 = π‘Š β†’ 𝑀 = π‘Š)
13 fveq2 5530 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
1413ineq2d 3351 . . . . 5 (𝑀 = π‘Š β†’ (π‘₯ ∩ (Baseβ€˜π‘€)) = (π‘₯ ∩ (Baseβ€˜π‘Š)))
1514opeq2d 3800 . . . 4 (𝑀 = π‘Š β†’ ⟨(Baseβ€˜ndx), (π‘₯ ∩ (Baseβ€˜π‘€))⟩ = ⟨(Baseβ€˜ndx), (π‘₯ ∩ (Baseβ€˜π‘Š))⟩)
1612, 15oveq12d 5909 . . 3 (𝑀 = π‘Š β†’ (𝑀 sSet ⟨(Baseβ€˜ndx), (π‘₯ ∩ (Baseβ€˜π‘€))⟩) = (π‘Š sSet ⟨(Baseβ€˜ndx), (π‘₯ ∩ (Baseβ€˜π‘Š))⟩))
17 ineq1 3344 . . . . 5 (π‘₯ = 𝐴 β†’ (π‘₯ ∩ (Baseβ€˜π‘Š)) = (𝐴 ∩ (Baseβ€˜π‘Š)))
1817opeq2d 3800 . . . 4 (π‘₯ = 𝐴 β†’ ⟨(Baseβ€˜ndx), (π‘₯ ∩ (Baseβ€˜π‘Š))⟩ = ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩)
1918oveq2d 5907 . . 3 (π‘₯ = 𝐴 β†’ (π‘Š sSet ⟨(Baseβ€˜ndx), (π‘₯ ∩ (Baseβ€˜π‘Š))⟩) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))
20 df-iress 12494 . . 3 β†Ύs = (𝑀 ∈ V, π‘₯ ∈ V ↦ (𝑀 sSet ⟨(Baseβ€˜ndx), (π‘₯ ∩ (Baseβ€˜π‘€))⟩))
2116, 19, 20ovmpog 6026 . 2 ((π‘Š ∈ V ∧ 𝐴 ∈ V ∧ (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩) ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))
222, 4, 11, 21syl3anc 1249 1 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  Vcvv 2752   ∩ cin 3143  βŸ¨cop 3610  β€˜cfv 5231  (class class class)co 5891  β„•cn 8938  ndxcnx 12483   sSet csts 12484  Basecbs 12486   β†Ύs cress 12487
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-inn 8939  df-ndx 12489  df-slot 12490  df-base 12492  df-sets 12493  df-iress 12494
This theorem is referenced by:  ressex  12549  ressval2  12550  ressbasd  12551  strressid  12555  ressval3d  12556  resseqnbasd  12557  ressinbasd  12558  ressressg  12559  mgpress  13252
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