Theorem strnfvn 13102

Description: Value of a structure component extractor E. Normally, E is a defined constant symbol such as Base (df-base 13087) 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 13126. (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 13101	. 2 |- ( T. -> ( E ` S ) = ( S ` N ) )
7	6	mptru 1406	1 |- ( E ` S ) = ( S ` N )

Syntax hints:   = wceq 1397   T. wtru 1398   e. wcel 2202   _Vcvv 2802   ` cfv 5326   NNcn 9142  Slot cslot 13080

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-slot 13085

This theorem is referenced by:  ndxarg  13104  strsl0  13130  baseval  13134