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Theorem strfvn 16484
Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 16468) 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 17535) 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 16510. (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 16481 . 2 (⊤ → (𝐸𝑆) = (𝑆𝑁))
54mptru 1545 1 (𝐸𝑆) = (𝑆𝑁)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wtru 1539  wcel 2115  Vcvv 3471  cfv 6328  Slot cslot 16461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-slot 16466
This theorem is referenced by:  ndxarg  16487  str0  16514  setsnid  16518  baseval  16521  ressbas  16533  resvsca  30911
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