Theorem basendx 13354

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 13353 and basendxnn 13355.

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 13464. Although we have a few theorems such as basendxnplusgndx 13425, 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 13305	. 2 ⊢ Base = Slot 1
2		1nn 9268	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 13322	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1398  ‘cfv 5357  1c1 8144  ndxcnx 13296  Basecbs 13299

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 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

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 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fv 5365  df-inn 9258  df-ndx 13302  df-slot 13303  df-base 13305

This theorem is referenced by:  basendxltplusgndx  13413  1strstrg  13416  2strstrg  13419  2strbasg  13420  2stropg  13421  2strstr1g  13422  rngstrg  13435  starvndxnbasendx  13442  scandxnbasendx  13454  vscandxnbasendx  13459  lmodstrd  13464  ipndxnbasendx  13472  basendxlttsetndx  13490  topgrpstrd  13496  basendxltplendx  13504  basendxnocndx  13513  basendxltdsndx  13519  basendxltunifndx  13529  setsmsbasg  15473  basendxltedgfndx  16134