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Theorem strnfvn 11994
Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 11979) 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 12017. (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 11993 . 2 (⊤ → (𝐸𝑆) = (𝑆𝑁))
76mptru 1340 1 (𝐸𝑆) = (𝑆𝑁)
Colors of variables: wff set class
Syntax hints:   = wceq 1331  wtru 1332  wcel 1480  Vcvv 2686  cfv 5123  cn 8732  Slot cslot 11972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-slot 11977
This theorem is referenced by:  ndxarg  11996  strsl0  12021  baseval  12025
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