Theorem islp 23254

Description: The predicate "the class 𝑃 is a limit point of 𝑆". (Contributed by NM, 10-Feb-2007.)

Hypothesis

Ref	Expression
lpfval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
islp	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))

Proof of Theorem islp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpfval.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	lpval 23253	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
3	2	eleq2d 2851	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))}))
4		id 23	. . 3 ⊢ (𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})) → 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})))
5		id 23	. . . 4 ⊢ (𝑥 = 𝑃 → 𝑥 = 𝑃)
6		sneq 4595	. . . . . 6 ⊢ (𝑥 = 𝑃 → {𝑥} = {𝑃})
7	6	difeq2d 4083	. . . . 5 ⊢ (𝑥 = 𝑃 → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑃}))
8	7	fveq2d 6875	. . . 4 ⊢ (𝑥 = 𝑃 → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) = ((cls‘𝐽)‘(𝑆 ∖ {𝑃})))
9	5, 8	eleq12d 2859	. . 3 ⊢ (𝑥 = 𝑃 → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))
10	4, 9	elab3 3648	. 2 ⊢ (𝑃 ∈ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})))
11	3, 10	bitrdi 290	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {cab 2743   ∖ cdif 3904   ⊆ wss 3907  {csn 4585  ∪ cuni 4867  ‘cfv 6525  Topctop 23007  clsccl 23132  limPtclp 23248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-top 23008  df-cld 23133  df-cls 23135  df-lp 23250

This theorem is referenced by:  lpdifsn  23257  lpss3  23258  islp2  23259  islp3  23260  maxlp  23261  restlp  23297  lpcls  23478  limcnlp  25994  limcflflem  25996