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

Theorem watfvalN 37770
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 3439 . 2 (𝐾𝐵𝐾 ∈ V)
2 watomfval.w . . 3 𝑊 = (WAtoms‘𝐾)
3 fveq2 6736 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 watomfval.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2797 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6736 . . . . . . 7 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
76fveq1d 6738 . . . . . 6 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘{𝑑}) = ((⊥𝑃𝐾)‘{𝑑}))
85, 7difeq12d 4053 . . . . 5 (𝑘 = 𝐾 → ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))
95, 8mpteq12dv 5155 . . . 4 (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑}))) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
10 df-watsN 37768 . . . 4 WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑}))))
119, 10, 4mptfvmpt 7063 . . 3 (𝐾 ∈ V → (WAtoms‘𝐾) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
122, 11syl5eq 2791 . 2 (𝐾 ∈ V → 𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
131, 12syl 17 1 (𝐾𝐵𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2111  Vcvv 3421  cdif 3878  {csn 4556  cmpt 5150  cfv 6398  Atomscatm 37041  𝑃cpolN 37680  WAtomscwpointsN 37764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5194  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-iun 4921  df-br 5069  df-opab 5131  df-mpt 5151  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406  df-watsN 37768
This theorem is referenced by:  watvalN  37771
  Copyright terms: Public domain W3C validator