Theorem strfvn 16989

Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 17015) 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 18133) 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 17007 instead of strfvn 16989. (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 16988	. 2 ⊢ (⊤ → (𝐸‘𝑆) = (𝑆‘𝑁))
5	4	mptru 1548	1 ⊢ (𝐸‘𝑆) = (𝑆‘𝑁)

Syntax hints:   = wceq 1541  ⊤wtru 1542   ∈ wcel 2106  Vcvv 3443  ‘cfv 6488  Slot cslot 16984

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5251  ax-nul 5258  ax-pr 5379

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4278  df-if 4482  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-br 5101  df-opab 5163  df-mpt 5184  df-id 5525  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-iota 6440  df-fun 6490  df-fv 6496  df-slot 16985

This theorem is referenced by:  str0  16992  ndxarg  16999  setsnid  17012  setsnidOLD  17013  baseval  17016  ressbasOLD  17050  resvsca  31889