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Theorem strfvn 17220
Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 17246) and 𝑁 is the index of the component. 𝑆 is a structure, i.e. a specific member of a class of structures such as Poset (df-poset 18371) where 𝑆 ∈ Poset.

Hint: Do not substitute 𝑁 by a specific (positive) integer to be independent of a hard-coded index value. Often, (𝐸‘ndx) can be used instead of 𝑁. Alternatively, use strfv 17238 instead of strfvn 17220. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.)

Hypotheses
Ref Expression
strfvn.f 𝑆 ∈ V
strfvn.c 𝐸 = Slot 𝑁
Assertion
Ref Expression
strfvn (𝐸𝑆) = (𝑆𝑁)

Proof of Theorem strfvn
StepHypRef Expression
1 strfvn.c . . 3 𝐸 = Slot 𝑁
2 strfvn.f . . . 4 𝑆 ∈ V
32a1i 11 . . 3 (⊤ → 𝑆 ∈ V)
41, 3strfvnd 17219 . 2 (⊤ → (𝐸𝑆) = (𝑆𝑁))
54mptru 1544 1 (𝐸𝑆) = (𝑆𝑁)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wtru 1538  wcel 2106  Vcvv 3478  cfv 6563  Slot cslot 17215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-slot 17216
This theorem is referenced by:  str0  17223  ndxarg  17230  setsnid  17243  setsnidOLD  17244  baseval  17247  ressbasOLD  17281  resvsca  33336
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