Theorem basendx 12676

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 12675 and basendxnn 12677.

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 12784. Although we have a few theorems such as basendxnplusgndx 12745, 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 12627	. 2 ⊢ Base = Slot 1
2		1nn 8995	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 12644	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1364  ‘cfv 5255  1c1 7875  ndxcnx 12618  Basecbs 12621

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fv 5263  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627

This theorem is referenced by:  basendxltplusgndx  12734  1strstrg  12737  2strstrg  12739  2strbasg  12740  2stropg  12741  2strstr1g  12742  rngstrg  12755  starvndxnbasendx  12762  scandxnbasendx  12774  vscandxnbasendx  12779  lmodstrd  12784  ipndxnbasendx  12792  basendxlttsetndx  12810  topgrpstrd  12816  basendxltplendx  12824  basendxltdsndx  12835  basendxltunifndx  12845  setsmsbasg  14658