watvalN - Mathbox for Norm Megill

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Theorem watvalN 40011

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 40010	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑}))))
5	4	fveq1d 6819	. 2 ⊢ (𝐾 ∈ 𝐵 → (𝑊‘𝐷) = ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))‘𝐷))
6		sneq 4584	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑑} = {𝐷})
7	6	fveq2d 6821	. . . 4 ⊢ (𝑑 = 𝐷 → ((⊥_𝑃‘𝐾)‘{𝑑}) = ((⊥_𝑃‘𝐾)‘{𝐷}))
8	7	difeq2d 4074	. . 3 ⊢ (𝑑 = 𝐷 → (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))
9		eqid 2730	. . 3 ⊢ (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))
10	1	fvexi 6831	. . . 4 ⊢ 𝐴 ∈ V
11	10	difexi 5266	. . 3 ⊢ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})) ∈ V
12	8, 9, 11	fvmpt 6924	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))
13	5, 12	sylan9eq 2785	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑊‘𝐷) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2110 ∖ cdif 3897 {csn 4574 ↦ cmpt 5170 ‘cfv 6477 Atomscatm 39281 ⊥_𝑃cpolN 39920 WAtomscwpointsN 40004

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pr 5368

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-watsN 40008

This theorem is referenced by: iswatN 40012

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