Theorem hmopex 28862

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.

Step	Hyp	Ref	Expression
1		ovex 6718	. 2 ⊢ ( ℋ ↑𝑚 ℋ) ∈ V
2		hmopf 28861	. . . 4 ⊢ (𝑡 ∈ HrmOp → 𝑡: ℋ⟶ ℋ)
3		ax-hilex 27984	. . . . 5 ⊢ ℋ ∈ V
4	3, 3	elmap 7928	. . . 4 ⊢ (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑡: ℋ⟶ ℋ)
5	2, 4	sylibr 224	. . 3 ⊢ (𝑡 ∈ HrmOp → 𝑡 ∈ ( ℋ ↑𝑚 ℋ))
6	5	ssriv 3640	. 2 ⊢ HrmOp ⊆ ( ℋ ↑𝑚 ℋ)
7	1, 6	ssexi 4836	1 ⊢ HrmOp ∈ V

Syntax hints:   ∈ wcel 2030  Vcvv 3231  ⟶wf 5922  (class class class)co 6690   ↑_𝑚 cmap 7899   ℋchil 27904  HrmOpcho 27935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-hilex 27984

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-hmop 28831

This theorem is referenced by: (None)