watvalN - Mathbox for Norm Megill

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Theorem watvalN 38312

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 38311	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑}))))
5	4	fveq1d 6836	. 2 ⊢ (𝐾 ∈ 𝐵 → (𝑊‘𝐷) = ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))‘𝐷))
6		sneq 4591	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑑} = {𝐷})
7	6	fveq2d 6838	. . . 4 ⊢ (𝑑 = 𝐷 → ((⊥_𝑃‘𝐾)‘{𝑑}) = ((⊥_𝑃‘𝐾)‘{𝐷}))
8	7	difeq2d 4077	. . 3 ⊢ (𝑑 = 𝐷 → (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))
9		eqid 2737	. . 3 ⊢ (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))
10	1	fvexi 6848	. . . 4 ⊢ 𝐴 ∈ V
11	10	difexi 5280	. . 3 ⊢ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})) ∈ V
12	8, 9, 11	fvmpt 6940	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))
13	5, 12	sylan9eq 2797	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑊‘𝐷) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1541 ∈ wcel 2106 ∖ cdif 3902 {csn 4581 ↦ cmpt 5183 ‘cfv 6488 Atomscatm 37581 ⊥_𝑃cpolN 38221 WAtomscwpointsN 38305

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2708 ax-rep 5237 ax-sep 5251 ax-nul 5258 ax-pr 5379

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3735 df-csb 3851 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4278 df-if 4482 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4861 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5184 df-id 5525 df-xp 5633 df-rel 5634 df-cnv 5635 df-co 5636 df-dm 5637 df-rn 5638 df-res 5639 df-ima 5640 df-iota 6440 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-watsN 38309

This theorem is referenced by: iswatN 38313

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