Theorem basendx 13267

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 13266 and basendxnn 13268.

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 13377. Although we have a few theorems such as basendxnplusgndx 13338, 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 13218	. 2 |- Base = Slot 1
2		1nn 9248	. 2 |- 1 e. NN
3	1, 2	ndxarg 13235	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1398   ` cfv 5352   1c1 8128   ndxcnx 13209   Basecbs 13212

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218

This theorem is referenced by:  basendxltplusgndx  13326  1strstrg  13329  2strstrg  13332  2strbasg  13333  2stropg  13334  2strstr1g  13335  rngstrg  13348  starvndxnbasendx  13355  scandxnbasendx  13367  vscandxnbasendx  13372  lmodstrd  13377  ipndxnbasendx  13385  basendxlttsetndx  13403  topgrpstrd  13409  basendxltplendx  13417  basendxnocndx  13426  basendxltdsndx  13432  basendxltunifndx  13442  setsmsbasg  15344  basendxltedgfndx  16005