Theorem watfvalN 38006

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 = (⊥𝑃‘𝐾)
watomfval.w	⊢ 𝑊 = (WAtoms‘𝐾)

Assertion

Ref	Expression
watfvalN	⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{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.

Step	Hyp	Ref	Expression
1		elex 3450	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		watomfval.w	. . 3 ⊢ 𝑊 = (WAtoms‘𝐾)
3		fveq2 6774	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		watomfval.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2796	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6774	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = (⊥𝑃‘𝐾))
7	6	fveq1d 6776	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘{𝑑}) = ((⊥𝑃‘𝐾)‘{𝑑}))
8	5, 7	difeq12d 4058	. . . . 5 ⊢ (𝑘 = 𝐾 → ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))
9	5, 8	mpteq12dv 5165	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))))
10		df-watsN 38004	. . . 4 ⊢ WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑}))))
11	9, 10, 4	mptfvmpt 7104	. . 3 ⊢ (𝐾 ∈ V → (WAtoms‘𝐾) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))))
12	2, 11	eqtrid 2790	. 2 ⊢ (𝐾 ∈ V → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))))
13	1, 12	syl 17	1 ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∖ cdif 3884  {csn 4561   ↦ cmpt 5157  ‘cfv 6433  Atomscatm 37277  ⊥_𝑃cpolN 37916  WAtomscwpointsN 38000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-watsN 38004

This theorem is referenced by:  watvalN  38007