Theorem watvalN 35597

Description: Value of 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
watvalN	⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑊‘𝐷) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷})))

Proof of Theorem watvalN

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		watomfval.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
2		watomfval.p	. . . 4 ⊢ 𝑃 = (⊥𝑃‘𝐾)
3		watomfval.w	. . . 4 ⊢ 𝑊 = (WAtoms‘𝐾)
4	1, 2, 3	watfvalN 35596	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))))
5	4	fveq1d 6231	. 2 ⊢ (𝐾 ∈ 𝐵 → (𝑊‘𝐷) = ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))‘𝐷))
6		sneq 4220	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑑} = {𝐷})
7	6	fveq2d 6233	. . . 4 ⊢ (𝑑 = 𝐷 → ((⊥𝑃‘𝐾)‘{𝑑}) = ((⊥𝑃‘𝐾)‘{𝐷}))
8	7	difeq2d 3761	. . 3 ⊢ (𝑑 = 𝐷 → (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷})))
9		eqid 2651	. . 3 ⊢ (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))
10		fvex 6239	. . . . 5 ⊢ (Atoms‘𝐾) ∈ V
11	1, 10	eqeltri 2726	. . . 4 ⊢ 𝐴 ∈ V
12		difexg 4841	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷})) ∈ V)
13	11, 12	ax-mp 5	. . 3 ⊢ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷})) ∈ V
14	8, 9, 13	fvmpt 6321	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷})))
15	5, 14	sylan9eq 2705	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑊‘𝐷) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷})))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604  {csn 4210   ↦ cmpt 4762  ‘cfv 5926  Atomscatm 34868  ⊥_𝑃cpolN 35506  WAtomscwpointsN 35590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-watsN 35594

This theorem is referenced by:  iswatN  35598