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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  ℕcn 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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