Theorem basendx 13053

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 13052 and basendxnn 13054.

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 13163. Although we have a few theorems such as basendxnplusgndx 13124, 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 13004	. 2 ⊢ Base = Slot 1
2		1nn 9089	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 13021	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1375  ‘cfv 5294  1c1 7968  ndxcnx 12995  Basecbs 12998

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-cnex 8058  ax-resscn 8059  ax-1re 8061  ax-addrcl 8064

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-sbc 3009  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-iota 5254  df-fun 5296  df-fv 5302  df-inn 9079  df-ndx 13001  df-slot 13002  df-base 13004

This theorem is referenced by:  basendxltplusgndx  13112  1strstrg  13115  2strstrg  13118  2strbasg  13119  2stropg  13120  2strstr1g  13121  rngstrg  13134  starvndxnbasendx  13141  scandxnbasendx  13153  vscandxnbasendx  13158  lmodstrd  13163  ipndxnbasendx  13171  basendxlttsetndx  13189  topgrpstrd  13195  basendxltplendx  13203  basendxnocndx  13212  basendxltdsndx  13218  basendxltunifndx  13228  setsmsbasg  15118  basendxltedgfndx  15776