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Theorem strnfvn 12453
Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 12438) 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 12476. (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 12452 . 2 (⊤ → (𝐸𝑆) = (𝑆𝑁))
76mptru 1362 1 (𝐸𝑆) = (𝑆𝑁)
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wtru 1354  wcel 2148  Vcvv 2737  cfv 5211  cn 8895  Slot cslot 12431
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fv 5219  df-slot 12436
This theorem is referenced by:  ndxarg  12455  strsl0  12480  baseval  12484
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