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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > watfvalN | Structured version Visualization version GIF version |
Description: The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
watomfval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
watomfval.p | ⊢ 𝑃 = (⊥𝑃‘𝐾) |
watomfval.w | ⊢ 𝑊 = (WAtoms‘𝐾) |
Ref | Expression |
---|---|
watfvalN | ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3361 | . 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V) | |
2 | watomfval.w | . . 3 ⊢ 𝑊 = (WAtoms‘𝐾) | |
3 | fveq2 6332 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾)) | |
4 | watomfval.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | syl6eqr 2822 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴) |
6 | fveq2 6332 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = (⊥𝑃‘𝐾)) | |
7 | 6 | fveq1d 6334 | . . . . . 6 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘{𝑑}) = ((⊥𝑃‘𝐾)‘{𝑑})) |
8 | 5, 7 | difeq12d 3878 | . . . . 5 ⊢ (𝑘 = 𝐾 → ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))) |
9 | 5, 8 | mpteq12dv 4865 | . . . 4 ⊢ (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))) |
10 | df-watsN 35791 | . . . 4 ⊢ WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑})))) | |
11 | fvex 6342 | . . . . . 6 ⊢ (Atoms‘𝐾) ∈ V | |
12 | 4, 11 | eqeltri 2845 | . . . . 5 ⊢ 𝐴 ∈ V |
13 | 12 | mptex 6629 | . . . 4 ⊢ (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))) ∈ V |
14 | 9, 10, 13 | fvmpt 6424 | . . 3 ⊢ (𝐾 ∈ V → (WAtoms‘𝐾) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))) |
15 | 2, 14 | syl5eq 2816 | . 2 ⊢ (𝐾 ∈ V → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))) |
16 | 1, 15 | syl 17 | 1 ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1630 ∈ wcel 2144 Vcvv 3349 ∖ cdif 3718 {csn 4314 ↦ cmpt 4861 ‘cfv 6031 Atomscatm 35065 ⊥𝑃cpolN 35703 WAtomscwpointsN 35787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-reu 3067 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-watsN 35791 |
This theorem is referenced by: watvalN 35794 |
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