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Theorem watfvalN 34095
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 3180 . 2 (𝐾𝐵𝐾 ∈ V)
2 watomfval.w . . 3 𝑊 = (WAtoms‘𝐾)
3 fveq2 6084 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 watomfval.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2657 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6084 . . . . . . 7 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
76fveq1d 6086 . . . . . 6 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘{𝑑}) = ((⊥𝑃𝐾)‘{𝑑}))
85, 7difeq12d 3686 . . . . 5 (𝑘 = 𝐾 → ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑})))
95, 8mpteq12dv 4653 . . . 4 (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑}))) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
10 df-watsN 34093 . . . 4 WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑}))))
11 fvex 6094 . . . . . 6 (Atoms‘𝐾) ∈ V
124, 11eqeltri 2679 . . . . 5 𝐴 ∈ V
1312mptex 6364 . . . 4 (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))) ∈ V
149, 10, 13fvmpt 6172 . . 3 (𝐾 ∈ V → (WAtoms‘𝐾) = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))
152, 14syl5eq 2651 . 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 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:  watvalN  34096
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