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Theorem watfvalN 35793
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 3361 . 2 (𝐾𝐵𝐾 ∈ V)
2 watomfval.w . . 3 𝑊 = (WAtoms‘𝐾)
3 fveq2 6332 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 watomfval.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2822 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6332 . . . . . . 7 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
76fveq1d 6334 . . . . . 6 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘{𝑑}) = ((⊥𝑃𝐾)‘{𝑑}))
85, 7difeq12d 3878 . . . . 5 (𝑘 = 𝐾 → ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))
95, 8mpteq12dv 4865 . . . 4 (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑}))) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
10 df-watsN 35791 . . . 4 WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑}))))
11 fvex 6342 . . . . . 6 (Atoms‘𝐾) ∈ V
124, 11eqeltri 2845 . . . . 5 𝐴 ∈ V
1312mptex 6629 . . . 4 (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))) ∈ V
149, 10, 13fvmpt 6424 . . 3 (𝐾 ∈ V → (WAtoms‘𝐾) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
152, 14syl5eq 2816 . 2 (𝐾 ∈ V → 𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
161, 15syl 17 1 (𝐾𝐵𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  Vcvv 3349  cdif 3718  {csn 4314  cmpt 4861  cfv 6031  Atomscatm 35065  𝑃cpolN 35703  WAtomscwpointsN 35787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-watsN 35791
This theorem is referenced by:  watvalN  35794
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