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Theorem hmopex 27513
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 6329 . 2  |-  ( ~H 
^m  ~H )  e.  _V
2 hmopf 27512 . . . 4  |-  ( t  e.  HrmOp  ->  t : ~H
--> ~H )
3 ax-hilex 26637 . . . . 5  |-  ~H  e.  _V
43, 3elmap 7504 . . . 4  |-  ( t  e.  ( ~H  ^m  ~H )  <->  t : ~H --> ~H )
52, 4sylibr 215 . . 3  |-  ( t  e.  HrmOp  ->  t  e.  ( ~H  ^m  ~H )
)
65ssriv 3468 . 2  |-  HrmOp  C_  ( ~H  ^m  ~H )
71, 6ssexi 4565 1  |-  HrmOp  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1868   _Vcvv 3081   -->wf 5593  (class class class)co 6301    ^m cmap 7476   ~Hchil 26557   HrmOpcho 26588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-hilex 26637
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-map 7478  df-hmop 27482
This theorem is referenced by: (None)
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