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Theorem hmopex 25230
Description: The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hmopex  |-  HrmOp  e.  _V

Proof of Theorem hmopex
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 ovex 6111 . 2  |-  ( ~H 
^m  ~H )  e.  _V
2 hmopf 25229 . . . 4  |-  ( t  e.  HrmOp  ->  t : ~H
--> ~H )
3 ax-hilex 24352 . . . . 5  |-  ~H  e.  _V
43, 3elmap 7233 . . . 4  |-  ( t  e.  ( ~H  ^m  ~H )  <->  t : ~H --> ~H )
52, 4sylibr 212 . . 3  |-  ( t  e.  HrmOp  ->  t  e.  ( ~H  ^m  ~H )
)
65ssriv 3355 . 2  |-  HrmOp  C_  ( ~H  ^m  ~H )
71, 6ssexi 4432 1  |-  HrmOp  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1756   _Vcvv 2967   -->wf 5409  (class class class)co 6086    ^m cmap 7206   ~Hchil 24272   HrmOpcho 24303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-hilex 24352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-hmop 25199
This theorem is referenced by: (None)
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