watvalN - Mathbox for Norm Megill

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Theorem watvalN 40565

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 40564	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑}))))
5	4	fveq1d 6858	. 2 ⊢ (𝐾 ∈ 𝐵 → (𝑊‘𝐷) = ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))‘𝐷))
6		sneq 4586	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑑} = {𝐷})
7	6	fveq2d 6860	. . . 4 ⊢ (𝑑 = 𝐷 → ((⊥_𝑃‘𝐾)‘{𝑑}) = ((⊥_𝑃‘𝐾)‘{𝐷}))
8	7	difeq2d 4075	. . 3 ⊢ (𝑑 = 𝐷 → (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))
9		eqid 2756	. . 3 ⊢ (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))
10	1	fvexi 6870	. . . 4 ⊢ 𝐴 ∈ V
11	10	difexi 5280	. . 3 ⊢ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})) ∈ V
12	8, 9, 11	fvmpt 6964	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))
13	5, 12	sylan9eq 2811	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑊‘𝐷) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1554 ∈ wcel 2136 ∖ cdif 3896 {csn 4576 ↦ cmpt 5175 ‘cfv 6510 Atomscatm 39835 ⊥_𝑃cpolN 40474 WAtomscwpointsN 40558

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pr 5384

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-ral 3071 df-rex 3081 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4281 df-if 4475 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4945 df-br 5095 df-opab 5157 df-mpt 5176 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-watsN 40562

This theorem is referenced by: iswatN 40566

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