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Theorem watvalN 39975
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 39974 . . 3 (𝐾𝐵𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
54fveq1d 6908 . 2 (𝐾𝐵 → (𝑊𝐷) = ((𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))‘𝐷))
6 sneq 4640 . . . . 5 (𝑑 = 𝐷 → {𝑑} = {𝐷})
76fveq2d 6910 . . . 4 (𝑑 = 𝐷 → ((⊥𝑃𝐾)‘{𝑑}) = ((⊥𝑃𝐾)‘{𝐷}))
87difeq2d 4135 . . 3 (𝑑 = 𝐷 → (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
9 eqid 2734 . . 3 (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))
101fvexi 6920 . . . 4 𝐴 ∈ V
1110difexi 5335 . . 3 (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})) ∈ V
128, 9, 11fvmpt 7015 . 2 (𝐷𝐴 → ((𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
135, 12sylan9eq 2794 1 ((𝐾𝐵𝐷𝐴) → (𝑊𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  cdif 3959  {csn 4630  cmpt 5230  cfv 6562  Atomscatm 39244  𝑃cpolN 39884  WAtomscwpointsN 39968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-watsN 39972
This theorem is referenced by:  iswatN  39976
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