Theorem basendx 13095

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 13094 and basendxnn 13096.

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 13205. Although we have a few theorems such as basendxnplusgndx 13166, 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 13046	. 2 ⊢ Base = Slot 1
2		1nn 9129	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 13063	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1395  ‘cfv 5318  1c1 8008  ndxcnx 13037  Basecbs 13040

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 8098  ax-resscn 8099  ax-1re 8101  ax-addrcl 8104

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 9119  df-ndx 13043  df-slot 13044  df-base 13046

This theorem is referenced by:  basendxltplusgndx  13154  1strstrg  13157  2strstrg  13160  2strbasg  13161  2stropg  13162  2strstr1g  13163  rngstrg  13176  starvndxnbasendx  13183  scandxnbasendx  13195  vscandxnbasendx  13200  lmodstrd  13205  ipndxnbasendx  13213  basendxlttsetndx  13231  topgrpstrd  13237  basendxltplendx  13245  basendxnocndx  13254  basendxltdsndx  13260  basendxltunifndx  13270  setsmsbasg  15161  basendxltedgfndx  15819