HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hmopex Structured version   Visualization version   GIF version

Theorem hmopex 31805
Description: The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hmopex HrmOp ∈ V

Proof of Theorem hmopex
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 ovex 7449 . 2 ( ℋ ↑m ℋ) ∈ V
2 hmopf 31804 . . . 4 (𝑡 ∈ HrmOp → 𝑡: ℋ⟶ ℋ)
3 ax-hilex 30929 . . . . 5 ℋ ∈ V
43, 3elmap 8892 . . . 4 (𝑡 ∈ ( ℋ ↑m ℋ) ↔ 𝑡: ℋ⟶ ℋ)
52, 4sylibr 233 . . 3 (𝑡 ∈ HrmOp → 𝑡 ∈ ( ℋ ↑m ℋ))
65ssriv 3982 . 2 HrmOp ⊆ ( ℋ ↑m ℋ)
71, 6ssexi 5319 1 HrmOp ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2099  Vcvv 3462  wf 6542  (class class class)co 7416  m cmap 8847  chba 30849  HrmOpcho 30880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-hilex 30929
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8849  df-hmop 31774
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator