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Theorem watvalN 38669
Description: Value of the W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
watomfval.a 𝐴 = (Atomsβ€˜πΎ)
watomfval.p 𝑃 = (βŠ₯π‘ƒβ€˜πΎ)
watomfval.w π‘Š = (WAtomsβ€˜πΎ)
Assertion
Ref Expression
watvalN ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (π‘Šβ€˜π·) = (𝐴 βˆ– ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝐷})))

Proof of Theorem watvalN
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 watomfval.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
2 watomfval.p . . . 4 𝑃 = (βŠ₯π‘ƒβ€˜πΎ)
3 watomfval.w . . . 4 π‘Š = (WAtomsβ€˜πΎ)
41, 2, 3watfvalN 38668 . . 3 (𝐾 ∈ 𝐡 β†’ π‘Š = (𝑑 ∈ 𝐴 ↦ (𝐴 βˆ– ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝑑}))))
54fveq1d 6880 . 2 (𝐾 ∈ 𝐡 β†’ (π‘Šβ€˜π·) = ((𝑑 ∈ 𝐴 ↦ (𝐴 βˆ– ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝑑})))β€˜π·))
6 sneq 4632 . . . . 5 (𝑑 = 𝐷 β†’ {𝑑} = {𝐷})
76fveq2d 6882 . . . 4 (𝑑 = 𝐷 β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝑑}) = ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝐷}))
87difeq2d 4118 . . 3 (𝑑 = 𝐷 β†’ (𝐴 βˆ– ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝑑})) = (𝐴 βˆ– ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝐷})))
9 eqid 2731 . . 3 (𝑑 ∈ 𝐴 ↦ (𝐴 βˆ– ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 βˆ– ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝑑})))
101fvexi 6892 . . . 4 𝐴 ∈ V
1110difexi 5321 . . 3 (𝐴 βˆ– ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝐷})) ∈ V
128, 9, 11fvmpt 6984 . 2 (𝐷 ∈ 𝐴 β†’ ((𝑑 ∈ 𝐴 ↦ (𝐴 βˆ– ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝑑})))β€˜π·) = (𝐴 βˆ– ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝐷})))
135, 12sylan9eq 2791 1 ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (π‘Šβ€˜π·) = (𝐴 βˆ– ((βŠ₯π‘ƒβ€˜πΎ)β€˜{𝐷})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βˆ– cdif 3941  {csn 4622   ↦ cmpt 5224  β€˜cfv 6532  Atomscatm 37938  βŠ₯𝑃cpolN 38578  WAtomscwpointsN 38662
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-watsN 38666
This theorem is referenced by:  iswatN  38670
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