Theorem basendx 13465

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 13464 and basendxnn 13466.

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 13577. Although we have a few theorems such as basendxnplusgndx 13538, 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 13416	. 2 |- Base = Slot 1
2		1nn 9324	. 2 |- 1 e. NN
3	1, 2	ndxarg 13433	1 |- ( Base ` ndx ) = 1

This proof depends on syntax axioms:   = wceq 1402   ` cfv 5380   1c1 8186   ndxcnx 13407   Basecbs 13410

This proof depends on 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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fv 5388  df-inn 9314  df-ndx 13413  df-slot 13414  df-base 13416

This theorem is used by:  basendxltplusgndx  13526  1strstrg  13529  2strstrg  13532  2strbasg  13533  2stropg  13534  2strstr1g  13535  rngstrg  13548  starvndxnbasendx  13555  scandxnbasendx  13567  vscandxnbasendx  13572  lmodstrd  13577  ipndxnbasendx  13585  basendxlttsetndx  13603  topgrpstrd  13609  basendxltplendx  13617  basendxnocndx  13626  basendxltdsndx  13632  basendxltunifndx  13642  setsmsbasg  15705  basendxltedgfndx  16456