Theorem islp 23451

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 23450	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
3	2	eleq2d 2847	. 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 4594	. . . . . 6 ⊢ (𝑥 = 𝑃 → {𝑥} = {𝑃})
7	6	difeq2d 4074	. . . . 5 ⊢ (𝑥 = 𝑃 → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑃}))
8	7	fveq2d 6887	. . . 4 ⊢ (𝑥 = 𝑃 → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) = ((cls‘𝐽)‘(𝑆 ∖ {𝑃})))
9	5, 8	eleq12d 2855	. . 3 ⊢ (𝑥 = 𝑃 → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))
10	4, 9	elab3 3640	. 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}))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739   ∖ cdif 3896   ⊆ wss 3899  {csn 4584  ∪ cuni 4867  ‘cfv 6537  Topctop 23204  clsccl 23329  limPtclp 23445

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  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 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 23205  df-cld 23330  df-cls 23332  df-lp 23447

This theorem is used by:  lpdifsn  23454  lpss3  23455  islp2  23456  islp3  23457  maxlp  23458  restlp  23494  lpcls  23675  limcnlp  26191  limcflflem  26193