Theorem baseval 12960

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 12913	. 2 |- Base = Slot 1
3		1nn 9067	. 2 |- 1 e. NN
4	1, 2, 3	strnfvn 12928	1 |- ( Base ` K ) = ( K ` 1 )

Syntax hints:   = wceq 1373   e. wcel 2177   _Vcvv 2773   ` cfv 5280   1c1 7946   Basecbs 12907

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-1re 8039

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-iota 5241  df-fun 5282  df-fv 5288  df-inn 9057  df-slot 12911  df-base 12913

This theorem is referenced by: (None)