Theorem basendx 12758

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 12757 and basendxnn 12759.

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 12866. Although we have a few theorems such as basendxnplusgndx 12827, 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 12709	. 2 ⊢ Base = Slot 1
2		1nn 9018	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 12726	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1364  ‘cfv 5259  1c1 7897  ndxcnx 12700  Basecbs 12703

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 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

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 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fv 5267  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709

This theorem is referenced by:  basendxltplusgndx  12816  1strstrg  12819  2strstrg  12821  2strbasg  12822  2stropg  12823  2strstr1g  12824  rngstrg  12837  starvndxnbasendx  12844  scandxnbasendx  12856  vscandxnbasendx  12861  lmodstrd  12866  ipndxnbasendx  12874  basendxlttsetndx  12892  topgrpstrd  12898  basendxltplendx  12906  basendxnocndx  12915  basendxltdsndx  12921  basendxltunifndx  12931  setsmsbasg  14799