Theorem basendx 13155

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 13154 and basendxnn 13156.

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 13265. Although we have a few theorems such as basendxnplusgndx 13226, 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 13106	. 2 |- Base = Slot 1
2		1nn 9154	. 2 |- 1 e. NN
3	1, 2	ndxarg 13123	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1397   ` cfv 5326   1c1 8033   ndxcnx 13097   Basecbs 13100

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fv 5334  df-inn 9144  df-ndx 13103  df-slot 13104  df-base 13106

This theorem is referenced by:  basendxltplusgndx  13214  1strstrg  13217  2strstrg  13220  2strbasg  13221  2stropg  13222  2strstr1g  13223  rngstrg  13236  starvndxnbasendx  13243  scandxnbasendx  13255  vscandxnbasendx  13260  lmodstrd  13265  ipndxnbasendx  13273  basendxlttsetndx  13291  topgrpstrd  13297  basendxltplendx  13305  basendxnocndx  13314  basendxltdsndx  13320  basendxltunifndx  13330  setsmsbasg  15222  basendxltedgfndx  15880