watvalN - Mathbox for Norm Megill

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Theorem watvalN 41316

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 41315	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑}))))
5	4	fveq1d 6918	. 2 ⊢ (𝐾 ∈ 𝐵 → (𝑊‘𝐷) = ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))‘𝐷))
6		sneq 4594	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑑} = {𝐷})
7	6	fveq2d 6920	. . . 4 ⊢ (𝑑 = 𝐷 → ((⊥_𝑃‘𝐾)‘{𝑑}) = ((⊥_𝑃‘𝐾)‘{𝐷}))
8	7	difeq2d 4066	. . 3 ⊢ (𝑑 = 𝐷 → (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))
9		eqid 2763	. . 3 ⊢ (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))
10	1	fvexi 6930	. . . 4 ⊢ 𝐴 ∈ V
11	10	difexi 5299	. . 3 ⊢ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})) ∈ V
12	8, 9, 11	fvmpt 7024	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))
13	5, 12	sylan9eq 2818	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑊‘𝐷) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∖ cdif 4048 {csn 4588 ↦ cmpt 5192 ‘cfv 6566 Atomscatm 40586 ⊥_𝑃cpolN 41225 WAtomscwpointsN 41309

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pr 5398

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-watsN 41313

This theorem is used by: iswatN 41317

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