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Theorem watvalN 39995
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 39994 . . 3 (𝐾𝐵𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
54fveq1d 6908 . 2 (𝐾𝐵 → (𝑊𝐷) = ((𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))‘𝐷))
6 sneq 4636 . . . . 5 (𝑑 = 𝐷 → {𝑑} = {𝐷})
76fveq2d 6910 . . . 4 (𝑑 = 𝐷 → ((⊥𝑃𝐾)‘{𝑑}) = ((⊥𝑃𝐾)‘{𝐷}))
87difeq2d 4126 . . 3 (𝑑 = 𝐷 → (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
9 eqid 2737 . . 3 (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))
101fvexi 6920 . . . 4 𝐴 ∈ V
1110difexi 5330 . . 3 (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})) ∈ V
128, 9, 11fvmpt 7016 . 2 (𝐷𝐴 → ((𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
135, 12sylan9eq 2797 1 ((𝐾𝐵𝐷𝐴) → (𝑊𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cdif 3948  {csn 4626  cmpt 5225  cfv 6561  Atomscatm 39264  𝑃cpolN 39904  WAtomscwpointsN 39988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-watsN 39992
This theorem is referenced by:  iswatN  39996
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