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Theorem hmopex 28604
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 6638 . 2 ( ℋ ↑𝑚 ℋ) ∈ V
2 hmopf 28603 . . . 4 (𝑡 ∈ HrmOp → 𝑡: ℋ⟶ ℋ)
3 ax-hilex 27726 . . . . 5 ℋ ∈ V
43, 3elmap 7838 . . . 4 (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑡: ℋ⟶ ℋ)
52, 4sylibr 224 . . 3 (𝑡 ∈ HrmOp → 𝑡 ∈ ( ℋ ↑𝑚 ℋ))
65ssriv 3591 . 2 HrmOp ⊆ ( ℋ ↑𝑚 ℋ)
71, 6ssexi 4768 1 HrmOp ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  Vcvv 3189  wf 5848  (class class class)co 6610  𝑚 cmap 7809  chil 27646  HrmOpcho 27677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-hilex 27726
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-hmop 28573
This theorem is referenced by: (None)
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