Theorem strfvn 15879

Description: Value of a structure component extractor E. Normally, E is a defined constant symbol such as Base (df-base 15863) 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 16946) where S e. Poset.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strfv 15907. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
strfvn.f	|- S e. _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 e. _V
3	2	a1i 11	. . 3 |- ( T. -> S e. _V )
4	1, 3	strfvnd 15876	. 2 |- ( T. -> ( E ` S ) = ( S ` N ) )
5	4	trud 1493	1 |- ( E ` S ) = ( S ` N )

Syntax hints:   = wceq 1483   T. wtru 1484   e. wcel 1990   _Vcvv 3200   ` cfv 5888  Slot cslot 15856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861

This theorem is referenced by:  ndxarg  15882  str0  15911  setsnid  15915  baseval  15918  ressbas  15930  resvsca  29830