Theorem basendx 12931

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 12930 and basendxnn 12932.

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 13040. Although we have a few theorems such as basendxnplusgndx 13001, 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 12882	. 2 ⊢ Base = Slot 1
2		1nn 9054	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 12899	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1373  ‘cfv 5276  1c1 7933  ndxcnx 12873  Basecbs 12876

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 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

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 3000  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-iota 5237  df-fun 5278  df-fv 5284  df-inn 9044  df-ndx 12879  df-slot 12880  df-base 12882

This theorem is referenced by:  basendxltplusgndx  12989  1strstrg  12992  2strstrg  12995  2strbasg  12996  2stropg  12997  2strstr1g  12998  rngstrg  13011  starvndxnbasendx  13018  scandxnbasendx  13030  vscandxnbasendx  13035  lmodstrd  13040  ipndxnbasendx  13048  basendxlttsetndx  13066  topgrpstrd  13072  basendxltplendx  13080  basendxnocndx  13089  basendxltdsndx  13095  basendxltunifndx  13105  setsmsbasg  14995  basendxltedgfndx  15653