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Theorem lkrfval 29885
 Description: The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lkrfval.d Scalar
lkrfval.o
lkrfval.f LFnl
lkrfval.k LKer
Assertion
Ref Expression
lkrfval
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem lkrfval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2964 . 2
2 lkrfval.k . . 3 LKer
3 fveq2 5728 . . . . . 6 LFnl LFnl
4 lkrfval.f . . . . . 6 LFnl
53, 4syl6eqr 2486 . . . . 5 LFnl
6 fveq2 5728 . . . . . . . . . 10 Scalar Scalar
7 lkrfval.d . . . . . . . . . 10 Scalar
86, 7syl6eqr 2486 . . . . . . . . 9 Scalar
98fveq2d 5732 . . . . . . . 8 Scalar
10 lkrfval.o . . . . . . . 8
119, 10syl6eqr 2486 . . . . . . 7 Scalar
1211sneqd 3827 . . . . . 6 Scalar
1312imaeq2d 5203 . . . . 5 Scalar
145, 13mpteq12dv 4287 . . . 4 LFnl Scalar
15 df-lkr 29884 . . . 4 LKer LFnl Scalar
16 fvex 5742 . . . . . 6 LFnl
174, 16eqeltri 2506 . . . . 5
1817mptex 5966 . . . 4
1914, 15, 18fvmpt 5806 . . 3 LKer
202, 19syl5eq 2480 . 2
211, 20syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  cvv 2956  csn 3814   cmpt 4266  ccnv 4877  cima 4881  cfv 5454  Scalarcsca 13532  c0g 13723  LFnlclfn 29855  LKerclk 29883 This theorem is referenced by:  lkrval  29886 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-lkr 29884
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