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Theorem watfvalN 36013
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 3400 . 2 (𝐾𝐵𝐾 ∈ V)
2 watomfval.w . . 3 𝑊 = (WAtoms‘𝐾)
3 fveq2 6411 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 watomfval.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2851 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6411 . . . . . . 7 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
76fveq1d 6413 . . . . . 6 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘{𝑑}) = ((⊥𝑃𝐾)‘{𝑑}))
85, 7difeq12d 3927 . . . . 5 (𝑘 = 𝐾 → ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))
95, 8mpteq12dv 4926 . . . 4 (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑}))) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
10 df-watsN 36011 . . . 4 WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑}))))
119, 10, 4mptfvmpt 6719 . . 3 (𝐾 ∈ V → (WAtoms‘𝐾) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
122, 11syl5eq 2845 . 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 1653  wcel 2157  Vcvv 3385  cdif 3766  {csn 4368  cmpt 4922  cfv 6101  Atomscatm 35284  𝑃cpolN 35923  WAtomscwpointsN 36007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-watsN 36011
This theorem is referenced by:  watvalN  36014
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