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Theorem hmopex 22455
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 5883 . 2  |-  ( ~H 
^m  ~H )  e.  _V
2 hmopf 22454 . . . 4  |-  ( t  e.  HrmOp  ->  t : ~H
--> ~H )
3 ax-hilex 21579 . . . . 5  |-  ~H  e.  _V
43, 3elmap 6796 . . . 4  |-  ( t  e.  ( ~H  ^m  ~H )  <->  t : ~H --> ~H )
52, 4sylibr 203 . . 3  |-  ( t  e.  HrmOp  ->  t  e.  ( ~H  ^m  ~H )
)
65ssriv 3184 . 2  |-  HrmOp  C_  ( ~H  ^m  ~H )
71, 6ssexi 4159 1  |-  HrmOp  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   _Vcvv 2788   -->wf 5251  (class class class)co 5858    ^m cmap 6772   ~Hchil 21499   HrmOpcho 21530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-hmop 22424
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