Theorem basendx 13351

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 13350 and basendxnn 13352.

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 13461. Although we have a few theorems such as basendxnplusgndx 13422, 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 13302	. 2 |- Base = Slot 1
2		1nn 9265	. 2 |- 1 e. NN
3	1, 2	ndxarg 13319	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1398   ` cfv 5357   1c1 8144   ndxcnx 13293   Basecbs 13296

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fv 5365  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302

This theorem is referenced by:  basendxltplusgndx  13410  1strstrg  13413  2strstrg  13416  2strbasg  13417  2stropg  13418  2strstr1g  13419  rngstrg  13432  starvndxnbasendx  13439  scandxnbasendx  13451  vscandxnbasendx  13456  lmodstrd  13461  ipndxnbasendx  13469  basendxlttsetndx  13487  topgrpstrd  13493  basendxltplendx  13501  basendxnocndx  13510  basendxltdsndx  13516  basendxltunifndx  13526  setsmsbasg  15470  basendxltedgfndx  16131