Theorem strfvn 16815

Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 16841) 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 17946) where 𝑆 ∈ Poset.

Hint: Do not substitute 𝑁 by a specific (positive) integer to be independent of a hard-coded index value. Often, (𝐸‘ndx) can be used instead of 𝑁. Alternatively, use strfv 16833 instead of strfvn 16815. (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	⊢ (𝐸‘𝑆) = (𝑆‘𝑁)

Proof of Theorem strfvn

Step	Hyp	Ref	Expression
1		strfvn.c	. . 3 ⊢ 𝐸 = Slot 𝑁
2		strfvn.f	. . . 4 ⊢ 𝑆 ∈ V
3	2	a1i 11	. . 3 ⊢ (⊤ → 𝑆 ∈ V)
4	1, 3	strfvnd 16814	. 2 ⊢ (⊤ → (𝐸‘𝑆) = (𝑆‘𝑁))
5	4	mptru 1546	1 ⊢ (𝐸‘𝑆) = (𝑆‘𝑁)

Syntax hints:   = wceq 1539  ⊤wtru 1540   ∈ wcel 2108  Vcvv 3422  ‘cfv 6418  Slot cslot 16810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-slot 16811

This theorem is referenced by:  str0  16818  ndxarg  16825  setsnid  16838  setsnidOLD  16839  baseval  16842  ressbasOLD  16874  resvsca  31431