Theorem basendx 12520

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 12519 and basendxnn 12521.

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 12625. Although we have a few theorems such as basendxnplusgndx 12586, 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 12471	. 2 |- Base = Slot 1
2		1nn 8933	. 2 |- 1 e. NN
3	1, 2	ndxarg 12488	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1353   ` cfv 5218   1c1 7815   ndxcnx 12462   Basecbs 12465

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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911

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 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fv 5226  df-inn 8923  df-ndx 12468  df-slot 12469  df-base 12471

This theorem is referenced by:  basendxltplusgndx  12575  1strstrg  12578  2strstrg  12580  2strbasg  12581  2stropg  12582  2strstr1g  12583  rngstrg  12596  starvndxnbasendx  12603  scandxnbasendx  12615  vscandxnbasendx  12620  lmodstrd  12625  ipndxnbasendx  12633  basendxlttsetndx  12651  topgrpstrd  12657  basendxltplendx  12665  basendxltdsndx  12676  basendxltunifndx  12686  setsmsbasg  14119