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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > watvalN | Structured version Visualization version GIF version |
Description: Value of 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 |
---|---|
watvalN | ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑊‘𝐷) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | watomfval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | watomfval.p | . . . 4 ⊢ 𝑃 = (⊥𝑃‘𝐾) | |
3 | watomfval.w | . . . 4 ⊢ 𝑊 = (WAtoms‘𝐾) | |
4 | 1, 2, 3 | watfvalN 37288 | . . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))) |
5 | 4 | fveq1d 6647 | . 2 ⊢ (𝐾 ∈ 𝐵 → (𝑊‘𝐷) = ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))‘𝐷)) |
6 | sneq 4535 | . . . . 5 ⊢ (𝑑 = 𝐷 → {𝑑} = {𝐷}) | |
7 | 6 | fveq2d 6649 | . . . 4 ⊢ (𝑑 = 𝐷 → ((⊥𝑃‘𝐾)‘{𝑑}) = ((⊥𝑃‘𝐾)‘{𝐷})) |
8 | 7 | difeq2d 4050 | . . 3 ⊢ (𝑑 = 𝐷 → (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷}))) |
9 | eqid 2798 | . . 3 ⊢ (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))) | |
10 | 1 | fvexi 6659 | . . . 4 ⊢ 𝐴 ∈ V |
11 | 10 | difexi 5196 | . . 3 ⊢ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷})) ∈ V |
12 | 8, 9, 11 | fvmpt 6745 | . 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷}))) |
13 | 5, 12 | sylan9eq 2853 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑊‘𝐷) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 {csn 4525 ↦ cmpt 5110 ‘cfv 6324 Atomscatm 36559 ⊥𝑃cpolN 37198 WAtomscwpointsN 37282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-watsN 37286 |
This theorem is referenced by: iswatN 37290 |
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