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Theorem strfvn 17066
Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 17092) and 𝑁 is the index of the component. 𝑆 is a structure, i.e. a specific member of a class of structures such as Poset (df-poset 18210) where 𝑆 ∈ Poset.

Hint: Do not substitute 𝑁 by a specific (positive) integer to be independent of a hard-coded index value. Often, (πΈβ€˜ndx) can be used instead of 𝑁. Alternatively, use strfv 17084 instead of strfvn 17066. (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 (πΈβ€˜π‘†) = (π‘†β€˜π‘)

Proof of Theorem strfvn
StepHypRef Expression
1 strfvn.c . . 3 𝐸 = Slot 𝑁
2 strfvn.f . . . 4 𝑆 ∈ V
32a1i 11 . . 3 (⊀ β†’ 𝑆 ∈ V)
41, 3strfvnd 17065 . 2 (⊀ β†’ (πΈβ€˜π‘†) = (π‘†β€˜π‘))
54mptru 1549 1 (πΈβ€˜π‘†) = (π‘†β€˜π‘)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  βŠ€wtru 1543   ∈ wcel 2107  Vcvv 3447  β€˜cfv 6500  Slot cslot 17061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-slot 17062
This theorem is referenced by:  str0  17069  ndxarg  17076  setsnid  17089  setsnidOLD  17090  baseval  17093  ressbasOLD  17127  resvsca  32175
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