Theorem basendx 13160

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 13159 and basendxnn 13161.

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 13270. Although we have a few theorems such as basendxnplusgndx 13231, 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 13111	. 2 ⊢ Base = Slot 1
2		1nn 9159	. 2 ⊢ 1 ∈ ℕ
3	1, 2	ndxarg 13128	1 ⊢ (Base‘ndx) = 1

Syntax hints:   = wceq 1397  ‘cfv 5328  1c1 8038  ndxcnx 13102  Basecbs 13105

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-cnex 8128  ax-resscn 8129  ax-1re 8131  ax-addrcl 8134

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fv 5336  df-inn 9149  df-ndx 13108  df-slot 13109  df-base 13111

This theorem is referenced by:  basendxltplusgndx  13219  1strstrg  13222  2strstrg  13225  2strbasg  13226  2stropg  13227  2strstr1g  13228  rngstrg  13241  starvndxnbasendx  13248  scandxnbasendx  13260  vscandxnbasendx  13265  lmodstrd  13270  ipndxnbasendx  13278  basendxlttsetndx  13296  topgrpstrd  13302  basendxltplendx  13310  basendxnocndx  13319  basendxltdsndx  13325  basendxltunifndx  13335  setsmsbasg  15232  basendxltedgfndx  15890