Theorem hauscmp 23455

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 1149	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝑆 ⊆ 𝑋)
2		hauscmp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3		eqid 2761	. . . . . 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 1204	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝐽 ∈ Haus)
5		simpl2 1205	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝑆 ⊆ 𝑋)
6		simpl3 1206	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → (𝐽 ↾t 𝑆) ∈ Comp)
7		simpr 488	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → 𝑥 ∈ (𝑋 ∖ 𝑆))
8	2, 3, 4, 5, 6, 7	hauscmplem 23454	. . . . 5 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋 ∖ 𝑆)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋 ∖ 𝑆)))
9		haustop 23379	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
10	9	3ad2ant1 1145	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝐽 ∈ Top)
11		elssuni 4894	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽)
12	11, 2	sseqtrrdi 3975	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ 𝑋)
13	2	sscls 23104	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑋) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
14	10, 12, 13	syl2an 605	. . . . . . . . 9 ⊢ (((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
15		sstr2 3941	. . . . . . . . 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 621	. . . . . . 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 3174	. . . . . 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 484	. . . . 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 3153	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → ∀𝑥 ∈ (𝑋 ∖ 𝑆)∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ (𝑋 ∖ 𝑆)))
22		eltop2 23023	. . . 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 23075	. . 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 723	1 ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝑆 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  {crab 3413   ∖ cdif 3899   ⊆ wss 3902  ∪ cuni 4862  ‘cfv 6516  (class class class)co 7391   ↾_t crest 17440  Topctop 22941  Clsdccld 23064  clsccl 23066  Hauscha 23356  Compccmp 23434

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 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  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-pss 3922  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-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  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-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-1o 8431  df-2o 8432  df-en 8922  df-dom 8923  df-fin 8925  df-fi 9351  df-rest 17442  df-topgen 17463  df-top 22942  df-topon 22959  df-bases 22994  df-cld 23067  df-cls 23069  df-haus 23363  df-cmp 23435

This theorem is referenced by:  txkgen  23700  cmphaushmeo  23848  cnheibor  25005