Theorem hauscmp 22795

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 1137	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝑆 ⊆ 𝑋)
2		hauscmp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3		eqid 2731	. . . . . 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 1191	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝐽 ∈ Haus)
5		simpl2 1192	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝑆 ⊆ 𝑋)
6		simpl3 1193	. . . . . 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 22794	. . . . 5 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋 ∖ 𝑆)))
9		haustop 22719	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
10	9	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝐽 ∈ Top)
11		elssuni 4903	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽)
12	11, 2	sseqtrrdi 3998	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ 𝑋)
13	2	sscls 22444	. . . . . . . . . 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 3954	. . . . . . . . 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 3161	. . . . . 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 3139	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → ∀𝑥 ∈ (𝑋 ∖ 𝑆)∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑋 ∖ 𝑆)))
22		eltop2 22362	. . . 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 22415	. . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069  {crab 3405   ∖ cdif 3910   ⊆ wss 3913  ∪ cuni 4870  ‘cfv 6501  (class class class)co 7362   ↾_t crest 17316  Topctop 22279  Clsdccld 22404  clsccl 22406  Hauscha 22696  Compccmp 22774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-fin 8894  df-fi 9356  df-rest 17318  df-topgen 17339  df-top 22280  df-topon 22297  df-bases 22333  df-cld 22407  df-cls 22409  df-haus 22703  df-cmp 22775

This theorem is referenced by:  txkgen  23040  cmphaushmeo  23188  cnheibor  24355