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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 6577 | . 2 ⊢ ( ℋ ↑𝑚 ℋ) ∈ V | |
2 | hmopf 28117 | . . . 4 ⊢ (𝑡 ∈ HrmOp → 𝑡: ℋ⟶ ℋ) | |
3 | ax-hilex 27240 | . . . . 5 ⊢ ℋ ∈ V | |
4 | 3, 3 | elmap 7772 | . . . 4 ⊢ (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑡: ℋ⟶ ℋ) |
5 | 2, 4 | sylibr 223 | . . 3 ⊢ (𝑡 ∈ HrmOp → 𝑡 ∈ ( ℋ ↑𝑚 ℋ)) |
6 | 5 | ssriv 3572 | . 2 ⊢ HrmOp ⊆ ( ℋ ↑𝑚 ℋ) |
7 | 1, 6 | ssexi 4731 | 1 ⊢ HrmOp ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ⟶wf 5800 (class class class)co 6549 ↑𝑚 cmap 7744 ℋchil 27160 HrmOpcho 27191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-hilex 27240 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-hmop 28087 |
This theorem is referenced by: (None) |
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