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Theorem strnfvn 12496
Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 12481) 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 12520. (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 12495 . 2 (⊤ → (𝐸𝑆) = (𝑆𝑁))
76mptru 1372 1 (𝐸𝑆) = (𝑆𝑁)
Colors of variables: wff set class
Syntax hints:   = wceq 1363  wtru 1364  wcel 2158  Vcvv 2749  cfv 5228  cn 8932  Slot cslot 12474
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fv 5236  df-slot 12479
This theorem is referenced by:  ndxarg  12498  strsl0  12524  baseval  12528
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