Theorem basendx 16849

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 16843 and basendxnn 16850.

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 lmodstr 16961. Although we have a few theorems such as basendxnplusgndx 16918, 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 16841	. 2 ⊢ Base = Slot 1
2		1nn 11914	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 16825	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1539  ‘cfv 6418  1c1 10803  ndxcnx 16822  Basecbs 16840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-1cn 10860  ax-addcl 10862

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-nn 11904  df-slot 16811  df-ndx 16823  df-base 16841

This theorem is referenced by:  basendxnn  16850  1strstr  16855  2strstr  16860  2strstr1OLD  16864  resslemOLD  16878  basendxltplusgndx  16917  grpbasex  16927  grpplusgx  16928  basendxnmulrndx  16931  rngstr  16934  starvndxnbasendx  16940  scandxnbasendx  16952  vscandxnbasendx  16957  lmodstr  16961  ipndxnbasendx  16968  basendxlttsetndx  16990  topgrpstr  16995  basendxltplendx  17003  otpsstr  17009  basendxltdsndx  17019  basendxltunifndx  17028  slotsbhcdif  17044  oppcbasOLD  17346  rescbasOLD  17459  rescabs  17464  catstr  17590  odubas  17925  ipostr  18162  mgpressOLD  19651  cnfldfun  20522  thlbas  20813  indistpsx  22068  tuslemOLD  23327  setsmsbas  23536  slotsinbpsd  26707  slotslnbpsd  26708  trkgstr  26709  eengstr  27251  baseltedgf  27266