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Theorem watvalN 37247
 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 37246 . . 3 (𝐾𝐵𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
54fveq1d 6654 . 2 (𝐾𝐵 → (𝑊𝐷) = ((𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))‘𝐷))
6 sneq 4549 . . . . 5 (𝑑 = 𝐷 → {𝑑} = {𝐷})
76fveq2d 6656 . . . 4 (𝑑 = 𝐷 → ((⊥𝑃𝐾)‘{𝑑}) = ((⊥𝑃𝐾)‘{𝐷}))
87difeq2d 4074 . . 3 (𝑑 = 𝐷 → (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
9 eqid 2822 . . 3 (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))
101fvexi 6666 . . . 4 𝐴 ∈ V
1110difexi 5208 . . 3 (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})) ∈ V
128, 9, 11fvmpt 6750 . 2 (𝐷𝐴 → ((𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
135, 12sylan9eq 2877 1 ((𝐾𝐵𝐷𝐴) → (𝑊𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114   ∖ cdif 3905  {csn 4539   ↦ cmpt 5122  ‘cfv 6334  Atomscatm 36517  ⊥𝑃cpolN 37156  WAtomscwpointsN 37240 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-watsN 37244 This theorem is referenced by:  iswatN  37248
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