Theorem basendx 13456

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 13455 and basendxnn 13457.

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 13567. Although we have a few theorems such as basendxnplusgndx 13528, 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 13407	. 2 |- Base = Slot 1
2		1nn 9317	. 2 |- 1 e. NN
3	1, 2	ndxarg 13424	1 |- ( Base ` ndx ) = 1

This proof depends on syntax axioms:   = wceq 1402   ` cfv 5377   1c1 8180   ndxcnx 13398   Basecbs 13401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fv 5385  df-inn 9307  df-ndx 13404  df-slot 13405  df-base 13407

This theorem is used by:  basendxltplusgndx  13516  1strstrg  13519  2strstrg  13522  2strbasg  13523  2stropg  13524  2strstr1g  13525  rngstrg  13538  starvndxnbasendx  13545  scandxnbasendx  13557  vscandxnbasendx  13562  lmodstrd  13567  ipndxnbasendx  13575  basendxlttsetndx  13593  topgrpstrd  13599  basendxltplendx  13607  basendxnocndx  13616  basendxltdsndx  13622  basendxltunifndx  13632  setsmsbasg  15629  basendxltedgfndx  16349