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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 7449 | . 2 ⊢ ( ℋ ↑m ℋ) ∈ V | |
2 | hmopf 31804 | . . . 4 ⊢ (𝑡 ∈ HrmOp → 𝑡: ℋ⟶ ℋ) | |
3 | ax-hilex 30929 | . . . . 5 ⊢ ℋ ∈ V | |
4 | 3, 3 | elmap 8892 | . . . 4 ⊢ (𝑡 ∈ ( ℋ ↑m ℋ) ↔ 𝑡: ℋ⟶ ℋ) |
5 | 2, 4 | sylibr 233 | . . 3 ⊢ (𝑡 ∈ HrmOp → 𝑡 ∈ ( ℋ ↑m ℋ)) |
6 | 5 | ssriv 3982 | . 2 ⊢ HrmOp ⊆ ( ℋ ↑m ℋ) |
7 | 1, 6 | ssexi 5319 | 1 ⊢ HrmOp ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 Vcvv 3462 ⟶wf 6542 (class class class)co 7416 ↑m cmap 8847 ℋchba 30849 HrmOpcho 30880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-hilex 30929 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-map 8849 df-hmop 31774 |
This theorem is referenced by: (None) |
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