Theorem hmopex 29323

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 6956	. 2 ⊢ ( ℋ ↑𝑚 ℋ) ∈ V
2		hmopf 29322	. . . 4 ⊢ (𝑡 ∈ HrmOp → 𝑡: ℋ⟶ ℋ)
3		ax-hilex 28445	. . . . 5 ⊢ ℋ ∈ V
4	3, 3	elmap 8171	. . . 4 ⊢ (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑡: ℋ⟶ ℋ)
5	2, 4	sylibr 226	. . 3 ⊢ (𝑡 ∈ HrmOp → 𝑡 ∈ ( ℋ ↑𝑚 ℋ))
6	5	ssriv 3825	. 2 ⊢ HrmOp ⊆ ( ℋ ↑𝑚 ℋ)
7	1, 6	ssexi 5042	1 ⊢ HrmOp ∈ V

Syntax hints:   ∈ wcel 2107  Vcvv 3398  ⟶wf 6133  (class class class)co 6924   ↑_𝑚 cmap 8142   ℋchba 28365  HrmOpcho 28396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-hilex 28445

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-map 8144  df-hmop 29292

This theorem is referenced by: (None)