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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 39994 | . . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))) |
| 5 | 4 | fveq1d 6908 | . 2 ⊢ (𝐾 ∈ 𝐵 → (𝑊‘𝐷) = ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))‘𝐷)) |
| 6 | sneq 4636 | . . . . 5 ⊢ (𝑑 = 𝐷 → {𝑑} = {𝐷}) | |
| 7 | 6 | fveq2d 6910 | . . . 4 ⊢ (𝑑 = 𝐷 → ((⊥𝑃‘𝐾)‘{𝑑}) = ((⊥𝑃‘𝐾)‘{𝐷})) |
| 8 | 7 | difeq2d 4126 | . . 3 ⊢ (𝑑 = 𝐷 → (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷}))) |
| 9 | eqid 2737 | . . 3 ⊢ (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))) | |
| 10 | 1 | fvexi 6920 | . . . 4 ⊢ 𝐴 ∈ V |
| 11 | 10 | difexi 5330 | . . 3 ⊢ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷})) ∈ V |
| 12 | 8, 9, 11 | fvmpt 7016 | . 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷}))) |
| 13 | 5, 12 | sylan9eq 2797 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑊‘𝐷) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷}))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 {csn 4626 ↦ cmpt 5225 ‘cfv 6561 Atomscatm 39264 ⊥𝑃cpolN 39904 WAtomscwpointsN 39988 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-watsN 39992 |
| This theorem is referenced by: iswatN 39996 |
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