Theorem basendx 12962

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 12961 and basendxnn 12963.

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 13071. Although we have a few theorems such as basendxnplusgndx 13032, 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 12913	. 2 |- Base = Slot 1
2		1nn 9067	. 2 |- 1 e. NN
3	1, 2	ndxarg 12930	1 |- ( Base ` ndx ) = 1

Syntax hints:   = wceq 1373   ` cfv 5280   1c1 7946   ndxcnx 12904   Basecbs 12907

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-iota 5241  df-fun 5282  df-fv 5288  df-inn 9057  df-ndx 12910  df-slot 12911  df-base 12913

This theorem is referenced by:  basendxltplusgndx  13020  1strstrg  13023  2strstrg  13026  2strbasg  13027  2stropg  13028  2strstr1g  13029  rngstrg  13042  starvndxnbasendx  13049  scandxnbasendx  13061  vscandxnbasendx  13066  lmodstrd  13071  ipndxnbasendx  13079  basendxlttsetndx  13097  topgrpstrd  13103  basendxltplendx  13111  basendxnocndx  13120  basendxltdsndx  13126  basendxltunifndx  13136  setsmsbasg  15026  basendxltedgfndx  15684