Theorem basendx 12733

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 12732 and basendxnn 12734.

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 12841. Although we have a few theorems such as basendxnplusgndx 12802, 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 12684	. 2 |- Base = Slot 1
2		1nn 9001	. 2 |- 1 e. NN
3	1, 2	ndxarg 12701	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1364   ` cfv 5258   1c1 7880   ndxcnx 12675   Basecbs 12678

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fv 5266  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684

This theorem is referenced by:  basendxltplusgndx  12791  1strstrg  12794  2strstrg  12796  2strbasg  12797  2stropg  12798  2strstr1g  12799  rngstrg  12812  starvndxnbasendx  12819  scandxnbasendx  12831  vscandxnbasendx  12836  lmodstrd  12841  ipndxnbasendx  12849  basendxlttsetndx  12867  topgrpstrd  12873  basendxltplendx  12881  basendxltdsndx  12892  basendxltunifndx  12902  setsmsbasg  14715