Theorem basendx 13256

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 13255 and basendxnn 13257.

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 13366. Although we have a few theorems such as basendxnplusgndx 13327, 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 13207	. 2 ⊢ Base = Slot 1
2		1nn 9244	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 13224	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1398  ‘cfv 5351  1c1 8124  ndxcnx 13198  Basecbs 13201

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-cnex 8214  ax-resscn 8215  ax-1re 8217  ax-addrcl 8220

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fv 5359  df-inn 9234  df-ndx 13204  df-slot 13205  df-base 13207

This theorem is referenced by:  basendxltplusgndx  13315  1strstrg  13318  2strstrg  13321  2strbasg  13322  2stropg  13323  2strstr1g  13324  rngstrg  13337  starvndxnbasendx  13344  scandxnbasendx  13356  vscandxnbasendx  13361  lmodstrd  13366  ipndxnbasendx  13374  basendxlttsetndx  13392  topgrpstrd  13398  basendxltplendx  13406  basendxnocndx  13415  basendxltdsndx  13421  basendxltunifndx  13431  setsmsbasg  15331  basendxltedgfndx  15992