Theorem basendx 12511

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 12510 and basendxnn 12512.

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 12616. Although we have a few theorems such as basendxnplusgndx 12577, 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 12462	. 2 |- Base = Slot 1
2		1nn 8928	. 2 |- 1 e. NN
3	1, 2	ndxarg 12479	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1353   ` cfv 5216   1c1 7811   ndxcnx 12453   Basecbs 12456

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fv 5224  df-inn 8918  df-ndx 12459  df-slot 12460  df-base 12462

This theorem is referenced by:  basendxltplusgndx  12566  1strstrg  12569  2strstrg  12571  2strbasg  12572  2stropg  12573  2strstr1g  12574  rngstrg  12587  starvndxnbasendx  12594  scandxnbasendx  12606  vscandxnbasendx  12611  lmodstrd  12616  ipndxnbasendx  12624  basendxlttsetndx  12639  topgrpstrd  12645  basendxltplendx  12653  basendxltdsndx  12664  basendxltunifndx  12674  setsmsbasg  13872