Theorem basendx 13288

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 13287 and basendxnn 13289.

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 13398. Although we have a few theorems such as basendxnplusgndx 13359, 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 13239	. 2 ⊢ Base = Slot 1
2		1nn 9253	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 13256	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1398  ‘cfv 5354  1c1 8133  ndxcnx 13230  Basecbs 13233

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-cnex 8223  ax-resscn 8224  ax-1re 8226  ax-addrcl 8229

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fv 5362  df-inn 9243  df-ndx 13236  df-slot 13237  df-base 13239

This theorem is referenced by:  basendxltplusgndx  13347  1strstrg  13350  2strstrg  13353  2strbasg  13354  2stropg  13355  2strstr1g  13356  rngstrg  13369  starvndxnbasendx  13376  scandxnbasendx  13388  vscandxnbasendx  13393  lmodstrd  13398  ipndxnbasendx  13406  basendxlttsetndx  13424  topgrpstrd  13430  basendxltplendx  13438  basendxnocndx  13447  basendxltdsndx  13453  basendxltunifndx  13463  setsmsbasg  15393  basendxltedgfndx  16054