Theorem basendx 13390

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 13389 and basendxnn 13391.

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 13501. Although we have a few theorems such as basendxnplusgndx 13462, 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 13341	. 2 ⊢ Base = Slot 1
2		1nn 9298	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 13358	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1402  ‘cfv 5375  1c1 8174  ndxcnx 13332  Basecbs 13335

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fv 5383  df-inn 9288  df-ndx 13338  df-slot 13339  df-base 13341

This theorem is referenced by:  basendxltplusgndx  13450  1strstrg  13453  2strstrg  13456  2strbasg  13457  2stropg  13458  2strstr1g  13459  rngstrg  13472  starvndxnbasendx  13479  scandxnbasendx  13491  vscandxnbasendx  13496  lmodstrd  13501  ipndxnbasendx  13509  basendxlttsetndx  13527  topgrpstrd  13533  basendxltplendx  13541  basendxnocndx  13550  basendxltdsndx  13556  basendxltunifndx  13566  setsmsbasg  15563  basendxltedgfndx  16234