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Theorem hmopex 29664
 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 7183 . 2 ( ℋ ↑m ℋ) ∈ V
2 hmopf 29663 . . . 4 (𝑡 ∈ HrmOp → 𝑡: ℋ⟶ ℋ)
3 ax-hilex 28788 . . . . 5 ℋ ∈ V
43, 3elmap 8432 . . . 4 (𝑡 ∈ ( ℋ ↑m ℋ) ↔ 𝑡: ℋ⟶ ℋ)
52, 4sylibr 237 . . 3 (𝑡 ∈ HrmOp → 𝑡 ∈ ( ℋ ↑m ℋ))
65ssriv 3958 . 2 HrmOp ⊆ ( ℋ ↑m ℋ)
71, 6ssexi 5213 1 HrmOp ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2115  Vcvv 3481  ⟶wf 6340  (class class class)co 7150   ↑m cmap 8403   ℋchba 28708  HrmOpcho 28739 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456  ax-hilex 28788 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8405  df-hmop 29633 This theorem is referenced by: (None)
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