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Theorem strfvn 16086
Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 16070) and 𝑁 is a fixed integer such as 1. 𝑆 is a structure, i.e. a specific member of a class of structures such as Poset (df-poset 17154) where 𝑆 ∈ Poset.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strfv 16114. (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 16083 . 2 (⊤ → (𝐸𝑆) = (𝑆𝑁))
54trud 1641 1 (𝐸𝑆) = (𝑆𝑁)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wtru 1632  wcel 2145  Vcvv 3351  cfv 6030  Slot cslot 16063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-slot 16068
This theorem is referenced by:  ndxarg  16089  str0  16118  setsnid  16122  baseval  16125  ressbas  16137  resvsca  30170
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