Theorem baseval 12446

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 ∈ V

Assertion

Ref	Expression
baseval	⊢ (Base‘𝐾) = (𝐾‘1)

Proof of Theorem baseval

Step	Hyp	Ref	Expression
1		baseval.k	. 2 ⊢ 𝐾 ∈ V
2		df-base 12400	. 2 ⊢ Base = Slot 1
3		1nn 8868	. 2 ⊢ 1 ∈ ℕ
4	1, 2, 3	strnfvn 12415	1 ⊢ (Base‘𝐾) = (𝐾‘1)

Syntax hints:   = wceq 1343   ∈ wcel 2136  Vcvv 2726  ‘cfv 5188  1c1 7754  Basecbs 12394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-1re 7847

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196  df-inn 8858  df-slot 12398  df-base 12400

This theorem is referenced by: (None)