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Theorem strslfv 12476
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 12438). By virtue of ndxslid 12457, 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 12444 . . 3 (𝑆 Struct 𝑋𝑆 ∈ V)
42, 3mp1i 10 . 2 (𝐶𝑉𝑆 ∈ V)
52structfun 12450 . . 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 4224 . . . . 5 (((𝐸‘ndx) ∈ ℕ ∧ 𝐶𝑉) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
108, 9mpan 424 . . . 4 (𝐶𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
11 snssg 3725 . . . 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 12474 1 (𝐶𝑉𝐶 = (𝐸𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  Vcvv 2737  wss 3129  {csn 3591  cop 3594   class class class wbr 4000  ccnv 4621  Fun wfun 5205  cfv 5211  cn 8895   Struct cstr 12428  ndxcnx 12429  Slot cslot 12431
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fv 5219  df-struct 12434  df-slot 12436
This theorem is referenced by:  strslfv3  12477
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