Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  watvalN Structured version   Visualization version   GIF version

Theorem watvalN 34297
 Description: Value of 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
watvalN ((𝐾𝐵𝐷𝐴) → (𝑊𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))

Proof of Theorem watvalN
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 watomfval.a . . . 4 𝐴 = (Atoms‘𝐾)
2 watomfval.p . . . 4 𝑃 = (⊥𝑃𝐾)
3 watomfval.w . . . 4 𝑊 = (WAtoms‘𝐾)
41, 2, 3watfvalN 34296 . . 3 (𝐾𝐵𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
54fveq1d 6105 . 2 (𝐾𝐵 → (𝑊𝐷) = ((𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))‘𝐷))
6 sneq 4135 . . . . 5 (𝑑 = 𝐷 → {𝑑} = {𝐷})
76fveq2d 6107 . . . 4 (𝑑 = 𝐷 → ((⊥𝑃𝐾)‘{𝑑}) = ((⊥𝑃𝐾)‘{𝐷}))
87difeq2d 3690 . . 3 (𝑑 = 𝐷 → (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
9 eqid 2610 . . 3 (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))
10 fvex 6113 . . . . 5 (Atoms‘𝐾) ∈ V
111, 10eqeltri 2684 . . . 4 𝐴 ∈ V
12 difexg 4735 . . . 4 (𝐴 ∈ V → (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})) ∈ V)
1311, 12ax-mp 5 . . 3 (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})) ∈ V
148, 9, 13fvmpt 6191 . 2 (𝐷𝐴 → ((𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
155, 14sylan9eq 2664 1 ((𝐾𝐵𝐷𝐴) → (𝑊𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = 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:  iswatN  34298
 Copyright terms: Public domain W3C validator