Theorem watfvalN 35278

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 3212	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		watomfval.w	. . 3 ⊢ 𝑊 = (WAtoms‘𝐾)
3		fveq2 6191	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		watomfval.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	syl6eqr 2674	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6191	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = (⊥𝑃‘𝐾))
7	6	fveq1d 6193	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘{𝑑}) = ((⊥𝑃‘𝐾)‘{𝑑}))
8	5, 7	difeq12d 3729	. . . . 5 ⊢ (𝑘 = 𝐾 → ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑})) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑})))
9	5, 8	mpteq12dv 4733	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))))
10		df-watsN 35276	. . . 4 ⊢ WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑}))))
11		fvex 6201	. . . . . 6 ⊢ (Atoms‘𝐾) ∈ V
12	4, 11	eqeltri 2697	. . . . 5 ⊢ 𝐴 ∈ V
13	12	mptex 6486	. . . 4 ⊢ (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))) ∈ V
14	9, 10, 13	fvmpt 6282	. . 3 ⊢ (𝐾 ∈ V → (WAtoms‘𝐾) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))))
15	2, 14	syl5eq 2668	. 2 ⊢ (𝐾 ∈ V → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))))
16	1, 15	syl 17	1 ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571  {csn 4177   ↦ cmpt 4729  ‘cfv 5888  Atomscatm 34550  ⊥_𝑃cpolN 35188  WAtomscwpointsN 35272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-watsN 35276

This theorem is referenced by:  watvalN  35279