Theorem watfvalN 39971

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 3457	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		watomfval.w	. . 3 ⊢ 𝑊 = (WAtoms‘𝐾)
3		fveq2 6822	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		watomfval.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2782	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6822	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = (⊥𝑃‘𝐾))
7	6	fveq1d 6824	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘{𝑑}) = ((⊥𝑃‘𝐾)‘{𝑑}))
8	5, 7	difeq12d 4078	. . . . 5 ⊢ (𝑘 = 𝐾 → ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))
9	5, 8	mpteq12dv 5179	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))))
10		df-watsN 39969	. . . 4 ⊢ WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑}))))
11	9, 10, 4	mptfvmpt 7164	. . 3 ⊢ (𝐾 ∈ V → (WAtoms‘𝐾) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))))
12	2, 11	eqtrid 2776	. 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 1540   ∈ wcel 2109  Vcvv 3436   ∖ cdif 3900  {csn 4577   ↦ cmpt 5173  ‘cfv 6482  Atomscatm 39242  ⊥_𝑃cpolN 39881  WAtomscwpointsN 39965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-watsN 39969

This theorem is referenced by:  watvalN  39972