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Theorem ressval3d 12556
Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
Hypotheses
Ref Expression
ressval3d.r 𝑅 = (𝑆 β†Ύs 𝐴)
ressval3d.b 𝐡 = (Baseβ€˜π‘†)
ressval3d.e 𝐸 = (Baseβ€˜ndx)
ressval3d.s (πœ‘ β†’ 𝑆 ∈ 𝑉)
ressval3d.f (πœ‘ β†’ Fun 𝑆)
ressval3d.d (πœ‘ β†’ 𝐸 ∈ dom 𝑆)
ressval3d.u (πœ‘ β†’ 𝐴 βŠ† 𝐡)
Assertion
Ref Expression
ressval3d (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Proof of Theorem ressval3d
StepHypRef Expression
1 ressval3d.r . . 3 𝑅 = (𝑆 β†Ύs 𝐴)
2 ressval3d.s . . . 4 (πœ‘ β†’ 𝑆 ∈ 𝑉)
3 ressval3d.b . . . . . 6 𝐡 = (Baseβ€˜π‘†)
4 basfn 12544 . . . . . . 7 Base Fn V
52elexd 2765 . . . . . . 7 (πœ‘ β†’ 𝑆 ∈ V)
6 funfvex 5547 . . . . . . . 8 ((Fun Base ∧ 𝑆 ∈ dom Base) β†’ (Baseβ€˜π‘†) ∈ V)
76funfni 5331 . . . . . . 7 ((Base Fn V ∧ 𝑆 ∈ V) β†’ (Baseβ€˜π‘†) ∈ V)
84, 5, 7sylancr 414 . . . . . 6 (πœ‘ β†’ (Baseβ€˜π‘†) ∈ V)
93, 8eqeltrid 2276 . . . . 5 (πœ‘ β†’ 𝐡 ∈ V)
10 ressval3d.u . . . . 5 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
119, 10ssexd 4158 . . . 4 (πœ‘ β†’ 𝐴 ∈ V)
12 ressvalsets 12548 . . . 4 ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) β†’ (𝑆 β†Ύs 𝐴) = (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘†))⟩))
132, 11, 12syl2anc 411 . . 3 (πœ‘ β†’ (𝑆 β†Ύs 𝐴) = (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘†))⟩))
141, 13eqtrid 2234 . 2 (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘†))⟩))
15 ressval3d.e . . . . 5 𝐸 = (Baseβ€˜ndx)
1615a1i 9 . . . 4 (πœ‘ β†’ 𝐸 = (Baseβ€˜ndx))
17 df-ss 3157 . . . . . 6 (𝐴 βŠ† 𝐡 ↔ (𝐴 ∩ 𝐡) = 𝐴)
1810, 17sylib 122 . . . . 5 (πœ‘ β†’ (𝐴 ∩ 𝐡) = 𝐴)
193ineq2i 3348 . . . . 5 (𝐴 ∩ 𝐡) = (𝐴 ∩ (Baseβ€˜π‘†))
2018, 19eqtr3di 2237 . . . 4 (πœ‘ β†’ 𝐴 = (𝐴 ∩ (Baseβ€˜π‘†)))
2116, 20opeq12d 3801 . . 3 (πœ‘ β†’ ⟨𝐸, 𝐴⟩ = ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘†))⟩)
2221oveq2d 5907 . 2 (πœ‘ β†’ (𝑆 sSet ⟨𝐸, 𝐴⟩) = (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘†))⟩))
2314, 22eqtr4d 2225 1 (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   = wceq 1364   ∈ wcel 2160  Vcvv 2752   ∩ cin 3143   βŠ† wss 3144  βŸ¨cop 3610  dom cdm 4641  Fun wfun 5225   Fn wfn 5226  β€˜cfv 5231  (class class class)co 5891  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-fn 5234  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: (None)
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