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Theorem strslfv 12560
Description: Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 with a component extractor 𝐸 (such as the base set extractor df-base 12521). By virtue of ndxslid 12540, this can be done without having to refer to the hard-coded numeric index of 𝐸. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.)
Hypotheses
Ref Expression
strfv.s 𝑆 Struct 𝑋
strslfv.e (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
strfv.n {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆
Assertion
Ref Expression
strslfv (𝐶𝑉𝐶 = (𝐸𝑆))

Proof of Theorem strslfv
StepHypRef Expression
1 strslfv.e . 2 (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
2 strfv.s . . 3 𝑆 Struct 𝑋
3 structex 12527 . . 3 (𝑆 Struct 𝑋𝑆 ∈ V)
42, 3mp1i 10 . 2 (𝐶𝑉𝑆 ∈ V)
52structfun 12533 . . 3 Fun 𝑆
65a1i 9 . 2 (𝐶𝑉 → Fun 𝑆)
7 strfv.n . . 3 {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆
81simpri 113 . . . . 5 (𝐸‘ndx) ∈ ℕ
9 opexg 4246 . . . . 5 (((𝐸‘ndx) ∈ ℕ ∧ 𝐶𝑉) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
108, 9mpan 424 . . . 4 (𝐶𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
11 snssg 3741 . . . 4 (⟨(𝐸‘ndx), 𝐶⟩ ∈ V → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 ↔ {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆))
1210, 11syl 14 . . 3 (𝐶𝑉 → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 ↔ {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆))
137, 12mpbiri 168 . 2 (𝐶𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
14 id 19 . 2 (𝐶𝑉𝐶𝑉)
151, 4, 6, 13, 14strslfv2d 12558 1 (𝐶𝑉𝐶 = (𝐸𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  Vcvv 2752  wss 3144  {csn 3607  cop 3610   class class class wbr 4018  ccnv 4643  Fun wfun 5229  cfv 5235  cn 8950   Struct cstr 12511  ndxcnx 12512  Slot cslot 12514
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 4192  ax-pr 4227  ax-un 4451
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-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fv 5243  df-struct 12517  df-slot 12519
This theorem is referenced by:  strslfv3  12561  cnfldbas  13885  cnfldadd  13886  cnfldmul  13887  cnfldcj  13888
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