Theorem perfopn 23123

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 23094	. . . . . . 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 22855	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
6	3, 5	sylib 218	. . . . 5 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
7		elssuni 4913	. . . . . . 7 ⊢ (𝑌 ∈ 𝐽 → 𝑌 ⊆ ∪ 𝐽)
8	7	adantl 481	. . . . . 6 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝑌 ⊆ ∪ 𝐽)
9	8, 4	sseqtrrdi 4000	. . . . 5 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝑌 ⊆ 𝑋)
10		resttopon 23099	. . . . 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 2838	. . 3 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
13		topontop 22851	. . 3 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
14	12, 13	syl 17	. 2 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ Top)
15	9	sselda 3958	. . . . . 6 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
16	4	perfi 23093	. . . . . . 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 2826	. . . . . 6 ⊢ ({𝑥} ∈ 𝐾 ↔ {𝑥} ∈ (𝐽 ↾t 𝑌))
20		restopn2 23115	. . . . . . . . 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 3132	. . 3 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → ∀𝑥 ∈ 𝑌 ¬ {𝑥} ∈ 𝐾)
28		toponuni 22852	. . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
29	12, 28	syl 17	. . 3 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝑌 = ∪ 𝐾)
30	27, 29	raleqtrdv 3307	. 2 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → ∀𝑥 ∈ ∪ 𝐾 ¬ {𝑥} ∈ 𝐾)
31		eqid 2735	. . 3 ⊢ ∪ 𝐾 = ∪ 𝐾
32	31	isperf3 23091	. 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 3051   ⊆ wss 3926  {csn 4601  ∪ cuni 4883  ‘cfv 6531  (class class class)co 7405   ↾_t crest 17434  Topctop 22831  TopOnctopon 22848  Perfcperf 23073

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-en 8960  df-fin 8963  df-fi 9423  df-rest 17436  df-topgen 17457  df-top 22832  df-topon 22849  df-bases 22884  df-cld 22957  df-ntr 22958  df-cls 22959  df-lp 23074  df-perf 23075

This theorem is referenced by:  perfdvf  25856