Theorem perfopn 23193

Description: An open subset of a perfect space is perfect. (Contributed by Mario Carneiro, 25-Dec-2016.)

Hypotheses

Ref	Expression
restcls.1	⊢ 𝑋 = ∪ 𝐽
restcls.2	⊢ 𝐾 = (𝐽 ↾t 𝑌)

Assertion

Ref	Expression
perfopn	⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ Perf)

Proof of Theorem perfopn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		restcls.2	. . . 4 ⊢ 𝐾 = (𝐽 ↾t 𝑌)
2		perftop 23164	. . . . . . 7 ⊢ (𝐽 ∈ Perf → 𝐽 ∈ Top)
3	2	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐽 ∈ Top)
4		restcls.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
5	4	toptopon 22923	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
6	3, 5	sylib 218	. . . . 5 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
7		elssuni 4937	. . . . . . 7 ⊢ (𝑌 ∈ 𝐽 → 𝑌 ⊆ ∪ 𝐽)
8	7	adantl 481	. . . . . 6 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝑌 ⊆ ∪ 𝐽)
9	8, 4	sseqtrrdi 4025	. . . . 5 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝑌 ⊆ 𝑋)
10		resttopon 23169	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
11	6, 9, 10	syl2anc 584	. . . 4 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
12	1, 11	eqeltrid 2845	. . 3 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
13		topontop 22919	. . 3 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
14	12, 13	syl 17	. 2 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ Top)
15	9	sselda 3983	. . . . . 6 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
16	4	perfi 23163	. . . . . . 7 ⊢ ((𝐽 ∈ Perf ∧ 𝑥 ∈ 𝑋) → ¬ {𝑥} ∈ 𝐽)
17	16	adantlr 715	. . . . . 6 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑋) → ¬ {𝑥} ∈ 𝐽)
18	15, 17	syldan 591	. . . . 5 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → ¬ {𝑥} ∈ 𝐽)
19	1	eleq2i 2833	. . . . . 6 ⊢ ({𝑥} ∈ 𝐾 ↔ {𝑥} ∈ (𝐽 ↾t 𝑌))
20		restopn2 23185	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ 𝐽) → ({𝑥} ∈ (𝐽 ↾t 𝑌) ↔ ({𝑥} ∈ 𝐽 ∧ {𝑥} ⊆ 𝑌)))
21	2, 20	sylan 580	. . . . . . . 8 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → ({𝑥} ∈ (𝐽 ↾t 𝑌) ↔ ({𝑥} ∈ 𝐽 ∧ {𝑥} ⊆ 𝑌)))
22	21	adantr 480	. . . . . . 7 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → ({𝑥} ∈ (𝐽 ↾t 𝑌) ↔ ({𝑥} ∈ 𝐽 ∧ {𝑥} ⊆ 𝑌)))
23		simpl 482	. . . . . . 7 ⊢ (({𝑥} ∈ 𝐽 ∧ {𝑥} ⊆ 𝑌) → {𝑥} ∈ 𝐽)
24	22, 23	biimtrdi 253	. . . . . 6 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → ({𝑥} ∈ (𝐽 ↾t 𝑌) → {𝑥} ∈ 𝐽))
25	19, 24	biimtrid 242	. . . . 5 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → ({𝑥} ∈ 𝐾 → {𝑥} ∈ 𝐽))
26	18, 25	mtod 198	. . . 4 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → ¬ {𝑥} ∈ 𝐾)
27	26	ralrimiva 3146	. . 3 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → ∀𝑥 ∈ 𝑌 ¬ {𝑥} ∈ 𝐾)
28		toponuni 22920	. . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
29	12, 28	syl 17	. . 3 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝑌 = ∪ 𝐾)
30	27, 29	raleqtrdv 3328	. 2 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → ∀𝑥 ∈ ∪ 𝐾 ¬ {𝑥} ∈ 𝐾)
31		eqid 2737	. . 3 ⊢ ∪ 𝐾 = ∪ 𝐾
32	31	isperf3 23161	. 2 ⊢ (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐾 ¬ {𝑥} ∈ 𝐾))
33	14, 30, 32	sylanbrc 583	1 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ Perf)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  {csn 4626  ∪ cuni 4907  ‘cfv 6561  (class class class)co 7431   ↾_t crest 17465  Topctop 22899  TopOnctopon 22916  Perfcperf 23143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-lp 23144  df-perf 23145

This theorem is referenced by:  perfdvf  25938