Theorem baseval 12528

Description: Value of the base set extractor. (Normally it is preferred to work with ( Base ` ndx ) rather than the hard-coded 1 in order to make structure theorems portable. This is an example of how to obtain it when needed.) (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)

Hypothesis

Ref	Expression
baseval.k	|- K e. _V

Assertion

Ref	Expression
baseval	|- ( Base ` K ) = ( K ` 1 )

Proof of Theorem baseval

Step	Hyp	Ref	Expression
1		baseval.k	. 2 |- K e. _V
2		df-base 12481	. 2 |- Base = Slot 1
3		1nn 8943	. 2 |- 1 e. NN
4	1, 2, 3	strnfvn 12496	1 |- ( Base ` K ) = ( K ` 1 )

Syntax hints:   = wceq 1363   e. wcel 2158   _Vcvv 2749   ` cfv 5228   1c1 7825   Basecbs 12475

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-1re 7918

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fv 5236  df-inn 8933  df-slot 12479  df-base 12481

This theorem is referenced by: (None)