Theorem strnfvn 12642

Description: Value of a structure component extractor E. Normally, E is a defined constant symbol such as Base (df-base 12627) and N is a fixed integer such as 1. S is a structure, i.e. a specific member of a class of structures.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12666. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.)

Hypotheses

Ref	Expression
strnfvn.f	|- S e. _V
strnfvn.c	|- E = Slot N
strnfvn.n	|- N e. NN

Assertion

Ref	Expression
strnfvn	|- ( E ` S ) = ( S ` N )

Proof of Theorem strnfvn

Step	Hyp	Ref	Expression
1		strnfvn.c	. . 3 |- E = Slot N
2		strnfvn.f	. . . 4 |- S e. _V
3	2	a1i 9	. . 3 |- ( T. -> S e. _V )
4		strnfvn.n	. . . 4 |- N e. NN
5	4	a1i 9	. . 3 |- ( T. -> N e. NN )
6	1, 3, 5	strnfvnd 12641	. 2 |- ( T. -> ( E ` S ) = ( S ` N ) )
7	6	mptru 1373	1 |- ( E ` S ) = ( S ` N )

Syntax hints:   = wceq 1364   T. wtru 1365   e. wcel 2164   _Vcvv 2760   ` cfv 5255   NNcn 8984  Slot cslot 12620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fv 5263  df-slot 12625

This theorem is referenced by:  ndxarg  12644  strsl0  12670  baseval  12674