Theorem islp 23187

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 23186	. . 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 22	. . 3 ⊢ (𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})) → 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})))
5		id 22	. . . 4 ⊢ (𝑥 = 𝑃 → 𝑥 = 𝑃)
6		sneq 4589	. . . . . 6 ⊢ (𝑥 = 𝑃 → {𝑥} = {𝑃})
7	6	difeq2d 4078	. . . . 5 ⊢ (𝑥 = 𝑃 → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑃}))
8	7	fveq2d 6865	. . . 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 3644	. 2 ⊢ (𝑃 ∈ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})))
11	3, 10	bitrdi 289	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {cab 2739   ∖ cdif 3899   ⊆ wss 3902  {csn 4579  ∪ cuni 4862  ‘cfv 6515  Topctop 22940  clsccl 23065  limPtclp 23181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-top 22941  df-cld 23066  df-cls 23068  df-lp 23183

This theorem is referenced by:  lpdifsn  23190  lpss3  23191  islp2  23192  islp3  23193  maxlp  23194  restlp  23230  lpcls  23411  limcnlp  25927  limcflflem  25929