Theorem baseval 13405

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 13358	. 2 |- Base = Slot 1
3		1nn 9315	. 2 |- 1 e. NN
4	1, 2, 3	strnfvn 13373	1 |- ( Base ` K ) = ( K ` 1 )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209   _Vcvv 2821   ` cfv 5377   1c1 8180   Basecbs 13352

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-1re 8273

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fv 5385  df-inn 9305  df-slot 13356  df-base 13358

This theorem is used by: (None)