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Theorem hmopex 25102
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 6105 . 2  |-  ( ~H 
^m  ~H )  e.  _V
2 hmopf 25101 . . . 4  |-  ( t  e.  HrmOp  ->  t : ~H
--> ~H )
3 ax-hilex 24224 . . . . 5  |-  ~H  e.  _V
43, 3elmap 7229 . . . 4  |-  ( t  e.  ( ~H  ^m  ~H )  <->  t : ~H --> ~H )
52, 4sylibr 212 . . 3  |-  ( t  e.  HrmOp  ->  t  e.  ( ~H  ^m  ~H )
)
65ssriv 3348 . 2  |-  HrmOp  C_  ( ~H  ^m  ~H )
71, 6ssexi 4425 1  |-  HrmOp  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1755   _Vcvv 2962   -->wf 5402  (class class class)co 6080    ^m cmap 7202   ~Hchil 24144   HrmOpcho 24175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-hilex 24224
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-map 7204  df-hmop 25071
This theorem is referenced by: (None)
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