Theorem basendx 13136

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 13135 and basendxnn 13137.

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 13246. Although we have a few theorems such as basendxnplusgndx 13207, 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 13087	. 2 |- Base = Slot 1
2		1nn 9153	. 2 |- 1 e. NN
3	1, 2	ndxarg 13104	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1397   ` cfv 5326   1c1 8032   ndxcnx 13078   Basecbs 13081

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fv 5334  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087

This theorem is referenced by:  basendxltplusgndx  13195  1strstrg  13198  2strstrg  13201  2strbasg  13202  2stropg  13203  2strstr1g  13204  rngstrg  13217  starvndxnbasendx  13224  scandxnbasendx  13236  vscandxnbasendx  13241  lmodstrd  13246  ipndxnbasendx  13254  basendxlttsetndx  13272  topgrpstrd  13278  basendxltplendx  13286  basendxnocndx  13295  basendxltdsndx  13301  basendxltunifndx  13311  setsmsbasg  15202  basendxltedgfndx  15860