Theorem hauscmp 22009

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 1133	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝑆 ⊆ 𝑋)
2		hauscmp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3		eqid 2821	. . . . . 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 1187	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝐽 ∈ Haus)
5		simpl2 1188	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝑆 ⊆ 𝑋)
6		simpl3 1189	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → (𝐽 ↾t 𝑆) ∈ Comp)
7		simpr 487	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝑥 ∈ (𝑋 ∖ 𝑆))
8	2, 3, 4, 5, 6, 7	hauscmplem 22008	. . . . 5 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋 ∖ 𝑆)))
9		haustop 21933	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
10	9	3ad2ant1 1129	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝐽 ∈ Top)
11		elssuni 4860	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽)
12	11, 2	sseqtrrdi 4017	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ 𝑋)
13	2	sscls 21658	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑋) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
14	10, 12, 13	syl2an 597	. . . . . . . . 9 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
15		sstr2 3973	. . . . . . . . 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 613	. . . . . . 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 3274	. . . . . 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 483	. . . . 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 3182	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → ∀𝑥 ∈ (𝑋 ∖ 𝑆)∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑋 ∖ 𝑆)))
22		eltop2 21577	. . . 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 259	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → (𝑋 ∖ 𝑆) ∈ 𝐽)
25	2	iscld 21629	. . 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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  {crab 3142   ∖ cdif 3932   ⊆ wss 3935  ∪ cuni 4831  ‘cfv 6349  (class class class)co 7150   ↾_t crest 16688  Topctop 21495  Clsdccld 21618  clsccl 21620  Hauscha 21910  Compccmp 21988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-en 8504  df-dom 8505  df-fin 8507  df-fi 8869  df-rest 16690  df-topgen 16711  df-top 21496  df-topon 21513  df-bases 21548  df-cld 21621  df-cls 21623  df-haus 21917  df-cmp 21989

This theorem is referenced by:  txkgen  22254  cmphaushmeo  22402  cnheibor  23553