Theorem basendx 13200

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 13199 and basendxnn 13201.

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 13310. Although we have a few theorems such as basendxnplusgndx 13271, 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 13151	. 2 |- Base = Slot 1
2		1nn 9196	. 2 |- 1 e. NN
3	1, 2	ndxarg 13168	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1398   ` cfv 5333   1c1 8076   ndxcnx 13142   Basecbs 13145

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151

This theorem is referenced by:  basendxltplusgndx  13259  1strstrg  13262  2strstrg  13265  2strbasg  13266  2stropg  13267  2strstr1g  13268  rngstrg  13281  starvndxnbasendx  13288  scandxnbasendx  13300  vscandxnbasendx  13305  lmodstrd  13310  ipndxnbasendx  13318  basendxlttsetndx  13336  topgrpstrd  13342  basendxltplendx  13350  basendxnocndx  13359  basendxltdsndx  13365  basendxltunifndx  13375  setsmsbasg  15273  basendxltedgfndx  15934