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Theorem strslfv 11687
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 11649). By virtue of ndxslid 11668, 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 11655 . . 3 (𝑆 Struct 𝑋𝑆 ∈ V)
42, 3mp1i 10 . 2 (𝐶𝑉𝑆 ∈ V)
52structfun 11661 . . 3 Fun 𝑆
65a1i 9 . 2 (𝐶𝑉 → Fun 𝑆)
7 strfv.n . . 3 {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆
81simpri 112 . . . . 5 (𝐸‘ndx) ∈ ℕ
9 opexg 4079 . . . . 5 (((𝐸‘ndx) ∈ ℕ ∧ 𝐶𝑉) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
108, 9mpan 416 . . . 4 (𝐶𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
11 snssg 3595 . . . 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 167 . 2 (𝐶𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
14 id 19 . 2 (𝐶𝑉𝐶𝑉)
151, 4, 6, 13, 14strslfv2d 11685 1 (𝐶𝑉𝐶 = (𝐸𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1296  wcel 1445  Vcvv 2633  wss 3013  {csn 3466  cop 3469   class class class wbr 3867  ccnv 4466  Fun wfun 5043  cfv 5049  cn 8520   Struct cstr 11639  ndxcnx 11640  Slot cslot 11642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-iota 5014  df-fun 5051  df-fv 5057  df-struct 11645  df-slot 11647
This theorem is referenced by:  strslfv3  11688
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