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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 7191 | . 2 ⊢ ( ℋ ↑m ℋ) ∈ V | |
2 | hmopf 29653 | . . . 4 ⊢ (𝑡 ∈ HrmOp → 𝑡: ℋ⟶ ℋ) | |
3 | ax-hilex 28778 | . . . . 5 ⊢ ℋ ∈ V | |
4 | 3, 3 | elmap 8437 | . . . 4 ⊢ (𝑡 ∈ ( ℋ ↑m ℋ) ↔ 𝑡: ℋ⟶ ℋ) |
5 | 2, 4 | sylibr 236 | . . 3 ⊢ (𝑡 ∈ HrmOp → 𝑡 ∈ ( ℋ ↑m ℋ)) |
6 | 5 | ssriv 3973 | . 2 ⊢ HrmOp ⊆ ( ℋ ↑m ℋ) |
7 | 1, 6 | ssexi 5228 | 1 ⊢ HrmOp ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 Vcvv 3496 ⟶wf 6353 (class class class)co 7158 ↑m cmap 8408 ℋchba 28698 HrmOpcho 28729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-hilex 28778 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-hmop 29623 |
This theorem is referenced by: (None) |
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