Theorem strnfvn 12475

Description: Value of a structure component extractor E. Normally, E is a defined constant symbol such as Base (df-base 12460) 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 12499. (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 12474	. 2 |- ( T. -> ( E ` S ) = ( S ` N ) )
7	6	mptru 1362	1 |- ( E ` S ) = ( S ` N )

Syntax hints:   = wceq 1353   T. wtru 1354   e. wcel 2148   _Vcvv 2737   ` cfv 5215   NNcn 8915  Slot cslot 12453

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5177  df-fun 5217  df-fv 5223  df-slot 12458

This theorem is referenced by:  ndxarg  12477  strsl0  12503  baseval  12507