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Theorem watfvalN 34296
 Description: The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
watomfval.a 𝐴 = (Atoms‘𝐾)
watomfval.p 𝑃 = (⊥𝑃𝐾)
watomfval.w 𝑊 = (WAtoms‘𝐾)
Assertion
Ref Expression
watfvalN (𝐾𝐵𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
Distinct variable groups:   𝐴,𝑑   𝐾,𝑑
Allowed substitution hints:   𝐵(𝑑)   𝑃(𝑑)   𝑊(𝑑)

Proof of Theorem watfvalN
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝐾𝐵𝐾 ∈ V)
2 watomfval.w . . 3 𝑊 = (WAtoms‘𝐾)
3 fveq2 6103 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 watomfval.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2662 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6103 . . . . . . 7 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
76fveq1d 6105 . . . . . 6 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘{𝑑}) = ((⊥𝑃𝐾)‘{𝑑}))
85, 7difeq12d 3691 . . . . 5 (𝑘 = 𝐾 → ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))
95, 8mpteq12dv 4663 . . . 4 (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑}))) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
10 df-watsN 34294 . . . 4 WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑}))))
11 fvex 6113 . . . . . 6 (Atoms‘𝐾) ∈ V
124, 11eqeltri 2684 . . . . 5 𝐴 ∈ V
1312mptex 6390 . . . 4 (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))) ∈ V
149, 10, 13fvmpt 6191 . . 3 (𝐾 ∈ V → (WAtoms‘𝐾) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
152, 14syl5eq 2656 . 2 (𝐾 ∈ V → 𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
161, 15syl 17 1 (𝐾𝐵𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537  {csn 4125   ↦ cmpt 4643  ‘cfv 5804  Atomscatm 33568  ⊥𝑃cpolN 34206  WAtomscwpointsN 34290 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 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-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-watsN 34294 This theorem is referenced by:  watvalN  34297
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