Theorem basendx 13103

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 13102 and basendxnn 13104.

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 13213. Although we have a few theorems such as basendxnplusgndx 13174, 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 13054	. 2 ⊢ Base = Slot 1
2		1nn 9132	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 13071	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1395  ‘cfv 5318  1c1 8011  ndxcnx 13045  Basecbs 13048

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 4258  ax-pr 4293  ax-un 4524  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

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 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fv 5326  df-inn 9122  df-ndx 13051  df-slot 13052  df-base 13054

This theorem is referenced by:  basendxltplusgndx  13162  1strstrg  13165  2strstrg  13168  2strbasg  13169  2stropg  13170  2strstr1g  13171  rngstrg  13184  starvndxnbasendx  13191  scandxnbasendx  13203  vscandxnbasendx  13208  lmodstrd  13213  ipndxnbasendx  13221  basendxlttsetndx  13239  topgrpstrd  13245  basendxltplendx  13253  basendxnocndx  13262  basendxltdsndx  13268  basendxltunifndx  13278  setsmsbasg  15169  basendxltedgfndx  15827