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Theorem lcfls1lem 32394
 Description: Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)
Hypothesis
Ref Expression
lcfls1.c
Assertion
Ref Expression
lcfls1lem
Distinct variable groups:   ,   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem lcfls1lem
StepHypRef Expression
1 fveq2 5730 . . . . . . 7
21fveq2d 5734 . . . . . 6
32fveq2d 5734 . . . . 5
43, 1eqeq12d 2452 . . . 4
52sseq1d 3377 . . . 4
64, 5anbi12d 693 . . 3
7 lcfls1.c . . 3
86, 7elrab2 3096 . 2
9 3anass 941 . 2
108, 9bitr4i 245 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726  crab 2711   wss 3322  cfv 5456 This theorem is referenced by:  lcfls1N  32395  lcfls1c  32396  lclkrslem1  32397  lclkrslem2  32398  lclkrs  32399 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464
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