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Theorem strslfv 12017
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 11979). By virtue of ndxslid 11998, 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 11985 . . 3 (𝑆 Struct 𝑋𝑆 ∈ V)
42, 3mp1i 10 . 2 (𝐶𝑉𝑆 ∈ V)
52structfun 11991 . . 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 4150 . . . . 5 (((𝐸‘ndx) ∈ ℕ ∧ 𝐶𝑉) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
108, 9mpan 420 . . . 4 (𝐶𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
11 snssg 3656 . . . 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 12015 1 (𝐶𝑉𝐶 = (𝐸𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  Vcvv 2686  wss 3071  {csn 3527  cop 3530   class class class wbr 3929  ccnv 4538  Fun wfun 5117  cfv 5123  cn 8732   Struct cstr 11969  ndxcnx 11970  Slot cslot 11972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-struct 11975  df-slot 11977
This theorem is referenced by:  strslfv3  12018
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