Theorem basendx 13082

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 13081 and basendxnn 13083.

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 13192. Although we have a few theorems such as basendxnplusgndx 13153, 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 13033	. 2 |- Base = Slot 1
2		1nn 9117	. 2 |- 1 e. NN
3	1, 2	ndxarg 13050	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1395   ` cfv 5317   1c1 7996   ndxcnx 13024   Basecbs 13027

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fv 5325  df-inn 9107  df-ndx 13030  df-slot 13031  df-base 13033

This theorem is referenced by:  basendxltplusgndx  13141  1strstrg  13144  2strstrg  13147  2strbasg  13148  2stropg  13149  2strstr1g  13150  rngstrg  13163  starvndxnbasendx  13170  scandxnbasendx  13182  vscandxnbasendx  13187  lmodstrd  13192  ipndxnbasendx  13200  basendxlttsetndx  13218  topgrpstrd  13224  basendxltplendx  13232  basendxnocndx  13241  basendxltdsndx  13247  basendxltunifndx  13257  setsmsbasg  15147  basendxltedgfndx  15805