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Theorem strfvn 17223
Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 17248) 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 18359) 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 17240 instead of strfvn 17223. (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 17222 . 2 (⊤ → (𝐸𝑆) = (𝑆𝑁))
54mptru 1547 1 (𝐸𝑆) = (𝑆𝑁)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wtru 1541  wcel 2108  Vcvv 3480  cfv 6561  Slot cslot 17218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-slot 17219
This theorem is referenced by:  str0  17226  ndxarg  17233  setsnid  17245  setsnidOLD  17246  baseval  17249  ressbasOLD  17281  resvsca  33356
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