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Theorem watvalN 34096
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 34095 . . 3 (𝐾𝐵𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
54fveq1d 6086 . 2 (𝐾𝐵 → (𝑊𝐷) = ((𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))‘𝐷))
6 sneq 4130 . . . . 5 (𝑑 = 𝐷 → {𝑑} = {𝐷})
76fveq2d 6088 . . . 4 (𝑑 = 𝐷 → ((⊥𝑃𝐾)‘{𝑑}) = ((⊥𝑃𝐾)‘{𝐷}))
87difeq2d 3685 . . 3 (𝑑 = 𝐷 → (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
9 eqid 2605 . . 3 (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))
10 fvex 6094 . . . . 5 (Atoms‘𝐾) ∈ V
111, 10eqeltri 2679 . . . 4 𝐴 ∈ V
12 difexg 4726 . . . 4 (𝐴 ∈ V → (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})) ∈ V)
1311, 12ax-mp 5 . . 3 (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})) ∈ V
148, 9, 13fvmpt 6172 . 2 (𝐷𝐴 → ((𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
155, 14sylan9eq 2659 1 ((𝐾𝐵𝐷𝐴) → (𝑊𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  Vcvv 3168  cdif 3532  {csn 4120  cmpt 4633  cfv 5786  Atomscatm 33367  𝑃cpolN 34005  WAtomscwpointsN 34089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-watsN 34093
This theorem is referenced by:  iswatN  34097
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