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Mirrors > Home > HSE Home > Th. List > hmopex | Structured version Visualization version GIF version |
Description: The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hmopex | ⊢ HrmOp ∈ V |
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 |
Colors of variables: wff setvar class |
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) |
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