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Theorem hmopex 23331
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 6065 . 2  |-  ( ~H 
^m  ~H )  e.  _V
2 hmopf 23330 . . . 4  |-  ( t  e.  HrmOp  ->  t : ~H
--> ~H )
3 ax-hilex 22455 . . . . 5  |-  ~H  e.  _V
43, 3elmap 7001 . . . 4  |-  ( t  e.  ( ~H  ^m  ~H )  <->  t : ~H --> ~H )
52, 4sylibr 204 . . 3  |-  ( t  e.  HrmOp  ->  t  e.  ( ~H  ^m  ~H )
)
65ssriv 3312 . 2  |-  HrmOp  C_  ( ~H  ^m  ~H )
71, 6ssexi 4308 1  |-  HrmOp  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   _Vcvv 2916   -->wf 5409  (class class class)co 6040    ^m cmap 6977   ~Hchil 22375   HrmOpcho 22406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-hilex 22455
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-map 6979  df-hmop 23300
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