Theorem basendx 13385

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 13384 and basendxnn 13386.

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 13495. Although we have a few theorems such as basendxnplusgndx 13456, 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 13336	. 2 |- Base = Slot 1
2		1nn 9294	. 2 |- 1 e. NN
3	1, 2	ndxarg 13353	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1402   ` cfv 5372   1c1 8170   ndxcnx 13327   Basecbs 13330

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fv 5380  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336

This theorem is referenced by:  basendxltplusgndx  13444  1strstrg  13447  2strstrg  13450  2strbasg  13451  2stropg  13452  2strstr1g  13453  rngstrg  13466  starvndxnbasendx  13473  scandxnbasendx  13485  vscandxnbasendx  13490  lmodstrd  13495  ipndxnbasendx  13503  basendxlttsetndx  13521  topgrpstrd  13527  basendxltplendx  13535  basendxnocndx  13544  basendxltdsndx  13550  basendxltunifndx  13560  setsmsbasg  15503  basendxltedgfndx  16165