Theorem basendx 13130

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 13129 and basendxnn 13131.

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 13240. Although we have a few theorems such as basendxnplusgndx 13201, 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 13081	. 2 ⊢ Base = Slot 1
2		1nn 9147	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 13098	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1395  ‘cfv 5324  1c1 8026  ndxcnx 13072  Basecbs 13075

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8116  ax-resscn 8117  ax-1re 8119  ax-addrcl 8122

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fv 5332  df-inn 9137  df-ndx 13078  df-slot 13079  df-base 13081

This theorem is referenced by:  basendxltplusgndx  13189  1strstrg  13192  2strstrg  13195  2strbasg  13196  2stropg  13197  2strstr1g  13198  rngstrg  13211  starvndxnbasendx  13218  scandxnbasendx  13230  vscandxnbasendx  13235  lmodstrd  13240  ipndxnbasendx  13248  basendxlttsetndx  13266  topgrpstrd  13272  basendxltplendx  13280  basendxnocndx  13289  basendxltdsndx  13295  basendxltunifndx  13305  setsmsbasg  15196  basendxltedgfndx  15854