Theorem hauscmp 23355

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 1134	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝑆 ⊆ 𝑋)
2		hauscmp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3		eqid 2725	. . . . . 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 1188	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝐽 ∈ Haus)
5		simpl2 1189	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝑆 ⊆ 𝑋)
6		simpl3 1190	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → (𝐽 ↾t 𝑆) ∈ Comp)
7		simpr 483	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝑥 ∈ (𝑋 ∖ 𝑆))
8	2, 3, 4, 5, 6, 7	hauscmplem 23354	. . . . 5 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋 ∖ 𝑆)))
9		haustop 23279	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
10	9	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝐽 ∈ Top)
11		elssuni 4941	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽)
12	11, 2	sseqtrrdi 4028	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ 𝑋)
13	2	sscls 23004	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑋) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
14	10, 12, 13	syl2an 594	. . . . . . . . 9 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
15		sstr2 3983	. . . . . . . . 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 610	. . . . . . 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 3157	. . . . . 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 479	. . . . 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 3135	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → ∀𝑥 ∈ (𝑋 ∖ 𝑆)∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑋 ∖ 𝑆)))
22		eltop2 22922	. . . 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 22975	. . 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 711	1 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝑆 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3050  ∃wrex 3059  {crab 3418   ∖ cdif 3941   ⊆ wss 3944  ∪ cuni 4909  ‘cfv 6549  (class class class)co 7419   ↾_t crest 17405  Topctop 22839  Clsdccld 22964  clsccl 22966  Hauscha 23256  Compccmp 23334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-fin 8968  df-fi 9436  df-rest 17407  df-topgen 17428  df-top 22840  df-topon 22857  df-bases 22893  df-cld 22967  df-cls 22969  df-haus 23263  df-cmp 23335

This theorem is referenced by:  txkgen  23600  cmphaushmeo  23748  cnheibor  24925