Theorem basendx 13127

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 13126 and basendxnn 13128.

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 13237. Although we have a few theorems such as basendxnplusgndx 13198, 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 13078	. 2 |- Base = Slot 1
2		1nn 9144	. 2 |- 1 e. NN
3	1, 2	ndxarg 13095	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1395   ` cfv 5324   1c1 8023   ndxcnx 13069   Basecbs 13072

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fv 5332  df-inn 9134  df-ndx 13075  df-slot 13076  df-base 13078

This theorem is referenced by:  basendxltplusgndx  13186  1strstrg  13189  2strstrg  13192  2strbasg  13193  2stropg  13194  2strstr1g  13195  rngstrg  13208  starvndxnbasendx  13215  scandxnbasendx  13227  vscandxnbasendx  13232  lmodstrd  13237  ipndxnbasendx  13245  basendxlttsetndx  13263  topgrpstrd  13269  basendxltplendx  13277  basendxnocndx  13286  basendxltdsndx  13292  basendxltunifndx  13302  setsmsbasg  15193  basendxltedgfndx  15851