Theorem basendx 13108

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 13107 and basendxnn 13109.

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 13218. Although we have a few theorems such as basendxnplusgndx 13179, 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 13059	. 2 ⊢ Base = Slot 1
2		1nn 9137	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 13076	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1395  ‘cfv 5321  1c1 8016  ndxcnx 13050  Basecbs 13053

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 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-cnex 8106  ax-resscn 8107  ax-1re 8109  ax-addrcl 8112

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 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-iota 5281  df-fun 5323  df-fv 5329  df-inn 9127  df-ndx 13056  df-slot 13057  df-base 13059

This theorem is referenced by:  basendxltplusgndx  13167  1strstrg  13170  2strstrg  13173  2strbasg  13174  2stropg  13175  2strstr1g  13176  rngstrg  13189  starvndxnbasendx  13196  scandxnbasendx  13208  vscandxnbasendx  13213  lmodstrd  13218  ipndxnbasendx  13226  basendxlttsetndx  13244  topgrpstrd  13250  basendxltplendx  13258  basendxnocndx  13267  basendxltdsndx  13273  basendxltunifndx  13283  setsmsbasg  15174  basendxltedgfndx  15832