watvalN - Mathbox for Norm Megill

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Theorem watvalN 37986

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 37985	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑}))))
5	4	fveq1d 6770	. 2 ⊢ (𝐾 ∈ 𝐵 → (𝑊‘𝐷) = ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))‘𝐷))
6		sneq 4576	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑑} = {𝐷})
7	6	fveq2d 6772	. . . 4 ⊢ (𝑑 = 𝐷 → ((⊥_𝑃‘𝐾)‘{𝑑}) = ((⊥_𝑃‘𝐾)‘{𝐷}))
8	7	difeq2d 4061	. . 3 ⊢ (𝑑 = 𝐷 → (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))
9		eqid 2739	. . 3 ⊢ (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑}))) = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))
10	1	fvexi 6782	. . . 4 ⊢ 𝐴 ∈ V
11	10	difexi 5255	. . 3 ⊢ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})) ∈ V
12	8, 9, 11	fvmpt 6869	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝑑})))‘𝐷) = (𝐴 ∖ ((⊥_𝑃‘𝐾)‘{𝐷})))
13	5, 12	sylan9eq 2799	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 2109 ∖ cdif 3888 {csn 4566 ↦ cmpt 5161 ‘cfv 6430 Atomscatm 37256 ⊥_𝑃cpolN 37895 WAtomscwpointsN 37979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pr 5355

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-watsN 37983

This theorem is referenced by: iswatN 37987

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