Theorem strnfvn 12437

Description: Value of a structure component extractor E. Normally, E is a defined constant symbol such as Base (df-base 12422) 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 12460. (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 12436	. 2 |- ( T. -> ( E ` S ) = ( S ` N ) )
7	6	mptru 1357	1 |- ( E ` S ) = ( S ` N )

Syntax hints:   = wceq 1348   T. wtru 1349   e. wcel 2141   _Vcvv 2730   ` cfv 5198   NNcn 8878  Slot cslot 12415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-slot 12420

This theorem is referenced by:  ndxarg  12439  strsl0  12464  baseval  12468