Theorem basendx 12673

Description: Index value of the base set extractor.

Use of this theorem is discouraged since the particular value 1 for the index is an implementation detail. It is generally sufficient to work with ( Base ` ndx ) and use theorems such as baseid 12672 and basendxnn 12674.

The main circumstance in which it is necessary to look at indices directly is when showing that a set of indices are disjoint, in proofs such as lmodstrd 12781. Although we have a few theorems such as basendxnplusgndx 12742, we do not intend to add such theorems for every pair of indices (which would be quadradically many in the number of indices).

(New usage is discouraged.) (Contributed by Mario Carneiro, 2-Aug-2013.)

Assertion

Ref	Expression
basendx	|- ( Base ` ndx ) = 1

Proof of Theorem basendx

Step	Hyp	Ref	Expression
1		df-base 12624	. 2 |- Base = Slot 1
2		1nn 8993	. 2 |- 1 e. NN
3	1, 2	ndxarg 12641	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1364   ` cfv 5254   1c1 7873   ndxcnx 12615   Basecbs 12618

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 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

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 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fv 5262  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624

This theorem is referenced by:  basendxltplusgndx  12731  1strstrg  12734  2strstrg  12736  2strbasg  12737  2stropg  12738  2strstr1g  12739  rngstrg  12752  starvndxnbasendx  12759  scandxnbasendx  12771  vscandxnbasendx  12776  lmodstrd  12781  ipndxnbasendx  12789  basendxlttsetndx  12807  topgrpstrd  12813  basendxltplendx  12821  basendxltdsndx  12832  basendxltunifndx  12842  setsmsbasg  14647