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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 35278 | . . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))) |
5 | 4 | fveq1d 6193 | . 2 ⊢ (𝐾 ∈ 𝐵 → (𝑊‘𝐷) = ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))‘𝐷)) |
6 | sneq 4187 | . . . . 5 ⊢ (𝑑 = 𝐷 → {𝑑} = {𝐷}) | |
7 | 6 | fveq2d 6195 | . . . 4 ⊢ (𝑑 = 𝐷 → ((⊥𝑃‘𝐾)‘{𝑑}) = ((⊥𝑃‘𝐾)‘{𝐷})) |
8 | 7 | difeq2d 3728 | . . 3 ⊢ (𝑑 = 𝐷 → (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷}))) |
9 | eqid 2622 | . . 3 ⊢ (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))) | |
10 | fvex 6201 | . . . . 5 ⊢ (Atoms‘𝐾) ∈ V | |
11 | 1, 10 | eqeltri 2697 | . . . 4 ⊢ 𝐴 ∈ V |
12 | difexg 4808 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷})) ∈ V) | |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷})) ∈ V |
14 | 8, 9, 13 | fvmpt 6282 | . 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷}))) |
15 | 5, 14 | sylan9eq 2676 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑊‘𝐷) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 {csn 4177 ↦ cmpt 4729 ‘cfv 5888 Atomscatm 34550 ⊥𝑃cpolN 35188 WAtomscwpointsN 35272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-watsN 35276 |
This theorem is referenced by: iswatN 35280 |
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