Theorem basendx 12508

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 12507 and basendxnn 12509.

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 12613. Although we have a few theorems such as basendxnplusgndx 12574, 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 12459	. 2 ⊢ Base = Slot 1
2		1nn 8925	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 12476	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1353  ‘cfv 5214  1c1 7808  ndxcnx 12450  Basecbs 12453

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 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-cnex 7898  ax-resscn 7899  ax-1re 7901  ax-addrcl 7904

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 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5176  df-fun 5216  df-fv 5222  df-inn 8915  df-ndx 12456  df-slot 12457  df-base 12459

This theorem is referenced by:  basendxltplusgndx  12563  1strstrg  12566  2strstrg  12568  2strbasg  12569  2stropg  12570  2strstr1g  12571  rngstrg  12584  starvndxnbasendx  12591  scandxnbasendx  12603  vscandxnbasendx  12608  lmodstrd  12613  ipndxnbasendx  12621  basendxlttsetndx  12636  topgrpstrd  12642  basendxltplendx  12650  basendxltdsndx  12661  basendxltunifndx  12671  setsmsbasg  13841