Theorem hauscmp 22558

Description: A compact subspace of a T2 space is closed. (Contributed by Jeff Hankins, 16-Jan-2010.) (Proof shortened by Mario Carneiro, 14-Dec-2013.)

Hypothesis

Ref	Expression
hauscmp.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
hauscmp	⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝑆 ∈ (Clsd‘𝐽))

Proof of Theorem hauscmp

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1136	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝑆 ⊆ 𝑋)
2		hauscmp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3		eqid 2738	. . . . . 6 ⊢ {𝑦 ∈ 𝐽 ∣ ∃𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋 ∖ 𝑦))} = {𝑦 ∈ 𝐽 ∣ ∃𝑤 ∈ 𝐽 (𝑥 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋 ∖ 𝑦))}
4		simpl1 1190	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝐽 ∈ Haus)
5		simpl2 1191	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝑆 ⊆ 𝑋)
6		simpl3 1192	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → (𝐽 ↾t 𝑆) ∈ Comp)
7		simpr 485	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝑥 ∈ (𝑋 ∖ 𝑆))
8	2, 3, 4, 5, 6, 7	hauscmplem 22557	. . . . 5 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋 ∖ 𝑆)))
9		haustop 22482	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
10	9	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝐽 ∈ Top)
11		elssuni 4871	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽)
12	11, 2	sseqtrrdi 3972	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ 𝑋)
13	2	sscls 22207	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑋) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
14	10, 12, 13	syl2an 596	. . . . . . . . 9 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
15		sstr2 3928	. . . . . . . . 9 ⊢ (𝑧 ⊆ ((cls‘𝐽)‘𝑧) → (((cls‘𝐽)‘𝑧) ⊆ (𝑋 ∖ 𝑆) → 𝑧 ⊆ (𝑋 ∖ 𝑆)))
16	14, 15	syl 17	. . . . . . . 8 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑧 ∈ 𝐽) → (((cls‘𝐽)‘𝑧) ⊆ (𝑋 ∖ 𝑆) → 𝑧 ⊆ (𝑋 ∖ 𝑆)))
17	16	anim2d 612	. . . . . . 7 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑧 ∈ 𝐽) → ((𝑥 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋 ∖ 𝑆)) → (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑋 ∖ 𝑆))))
18	17	reximdva 3203	. . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → (∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋 ∖ 𝑆)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑋 ∖ 𝑆))))
19	18	adantr 481	. . . . 5 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → (∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋 ∖ 𝑆)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑋 ∖ 𝑆))))
20	8, 19	mpd 15	. . . 4 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑋 ∖ 𝑆)))
21	20	ralrimiva 3103	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → ∀𝑥 ∈ (𝑋 ∖ 𝑆)∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑋 ∖ 𝑆)))
22		eltop2 22125	. . . 4 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝑆) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋 ∖ 𝑆)∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑋 ∖ 𝑆))))
23	10, 22	syl 17	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → ((𝑋 ∖ 𝑆) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋 ∖ 𝑆)∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑋 ∖ 𝑆))))
24	21, 23	mpbird 256	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → (𝑋 ∖ 𝑆) ∈ 𝐽)
25	2	iscld 22178	. . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)))
26	10, 25	syl 17	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)))
27	1, 24, 26	mpbir2and 710	1 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝑆 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068   ∖ cdif 3884   ⊆ wss 3887  ∪ cuni 4839  ‘cfv 6433  (class class class)co 7275   ↾_t crest 17131  Topctop 22042  Clsdccld 22167  clsccl 22169  Hauscha 22459  Compccmp 22537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cld 22170  df-cls 22172  df-haus 22466  df-cmp 22538

This theorem is referenced by:  txkgen  22803  cmphaushmeo  22951  cnheibor  24118