Theorem strfvn 17210

Description: Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 17234) 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 18330) 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 17227 instead of strfvn 17210. (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 17209	. 2 ⊢ (⊤ → (𝐸‘𝑆) = (𝑆‘𝑁))
5	4	mptru 1547	1 ⊢ (𝐸‘𝑆) = (𝑆‘𝑁)

Syntax hints:   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  Vcvv 3464  ‘cfv 6536  Slot cslot 17205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-slot 17206

This theorem is referenced by:  str0  17213  ndxarg  17220  setsnid  17232  baseval  17235  resvsca  33353