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Theorem strnfvn 12501
Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 12486) and 𝑁 is a fixed integer such as 1. 𝑆 is a structure, i.e. a specific member of a class of structures.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12525. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.)

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

Proof of Theorem strnfvn
StepHypRef Expression
1 strnfvn.c . . 3 𝐸 = Slot 𝑁
2 strnfvn.f . . . 4 𝑆 ∈ V
32a1i 9 . . 3 (⊤ → 𝑆 ∈ V)
4 strnfvn.n . . . 4 𝑁 ∈ ℕ
54a1i 9 . . 3 (⊤ → 𝑁 ∈ ℕ)
61, 3, 5strnfvnd 12500 . 2 (⊤ → (𝐸𝑆) = (𝑆𝑁))
76mptru 1373 1 (𝐸𝑆) = (𝑆𝑁)
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wtru 1365  wcel 2160  Vcvv 2752  cfv 5231  cn 8937  Slot cslot 12479
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fv 5239  df-slot 12484
This theorem is referenced by:  ndxarg  12503  strsl0  12529  baseval  12533
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