Theorem basendx 12517

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 12516 and basendxnn 12518.

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 12622. Although we have a few theorems such as basendxnplusgndx 12583, 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 12468	. 2 ⊢ Base = Slot 1
2		1nn 8930	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 12485	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1353  ‘cfv 5217  1c1 7812  ndxcnx 12459  Basecbs 12462

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-iota 5179  df-fun 5219  df-fv 5225  df-inn 8920  df-ndx 12465  df-slot 12466  df-base 12468

This theorem is referenced by:  basendxltplusgndx  12572  1strstrg  12575  2strstrg  12577  2strbasg  12578  2stropg  12579  2strstr1g  12580  rngstrg  12593  starvndxnbasendx  12600  scandxnbasendx  12612  vscandxnbasendx  12617  lmodstrd  12622  ipndxnbasendx  12630  basendxlttsetndx  12645  topgrpstrd  12651  basendxltplendx  12659  basendxltdsndx  12670  basendxltunifndx  12680  setsmsbasg  13982