Theorem perfopn 22434

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 22405	. . . . . . 7 ⊢ (𝐽 ∈ Perf → 𝐽 ∈ Top)
3	2	adantr 481	. . . . . 6 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐽 ∈ Top)
4		restcls.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
5	4	toptopon 22164	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
6	3, 5	sylib 217	. . . . 5 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
7		elssuni 4884	. . . . . . 7 ⊢ (𝑌 ∈ 𝐽 → 𝑌 ⊆ ∪ 𝐽)
8	7	adantl 482	. . . . . 6 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝑌 ⊆ ∪ 𝐽)
9	8, 4	sseqtrrdi 3982	. . . . 5 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝑌 ⊆ 𝑋)
10		resttopon 22410	. . . . 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 2841	. . 3 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
13		topontop 22160	. . 3 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
14	12, 13	syl 17	. 2 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ Top)
15	9	sselda 3931	. . . . . 6 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
16	4	perfi 22404	. . . . . . 7 ⊢ ((𝐽 ∈ Perf ∧ 𝑥 ∈ 𝑋) → ¬ {𝑥} ∈ 𝐽)
17	16	adantlr 712	. . . . . 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 2828	. . . . . 6 ⊢ ({𝑥} ∈ 𝐾 ↔ {𝑥} ∈ (𝐽 ↾t 𝑌))
20		restopn2 22426	. . . . . . . . 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 481	. . . . . . 7 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → ({𝑥} ∈ (𝐽 ↾t 𝑌) ↔ ({𝑥} ∈ 𝐽 ∧ {𝑥} ⊆ 𝑌)))
23		simpl 483	. . . . . . 7 ⊢ (({𝑥} ∈ 𝐽 ∧ {𝑥} ⊆ 𝑌) → {𝑥} ∈ 𝐽)
24	22, 23	syl6bi 252	. . . . . 6 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → ({𝑥} ∈ (𝐽 ↾t 𝑌) → {𝑥} ∈ 𝐽))
25	19, 24	biimtrid 241	. . . . 5 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → ({𝑥} ∈ 𝐾 → {𝑥} ∈ 𝐽))
26	18, 25	mtod 197	. . . 4 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → ¬ {𝑥} ∈ 𝐾)
27	26	ralrimiva 3139	. . 3 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → ∀𝑥 ∈ 𝑌 ¬ {𝑥} ∈ 𝐾)
28		toponuni 22161	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
29	12, 28	syl 17	. . . 4 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝑌 = ∪ 𝐾)
30	29	raleqdv 3309	. . 3 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → (∀𝑥 ∈ 𝑌 ¬ {𝑥} ∈ 𝐾 ↔ ∀𝑥 ∈ ∪ 𝐾 ¬ {𝑥} ∈ 𝐾))
31	27, 30	mpbid 231	. 2 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → ∀𝑥 ∈ ∪ 𝐾 ¬ {𝑥} ∈ 𝐾)
32		eqid 2736	. . 3 ⊢ ∪ 𝐾 = ∪ 𝐾
33	32	isperf3 22402	. 2 ⊢ (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐾 ¬ {𝑥} ∈ 𝐾))
34	14, 31, 33	sylanbrc 583	1 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ Perf)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3061   ⊆ wss 3897  {csn 4572  ∪ cuni 4851  ‘cfv 6473  (class class class)co 7329   ↾_t crest 17220  Topctop 22140  TopOnctopon 22157  Perfcperf 22384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-iin 4941  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-en 8797  df-fin 8800  df-fi 9260  df-rest 17222  df-topgen 17243  df-top 22141  df-topon 22158  df-bases 22194  df-cld 22268  df-ntr 22269  df-cls 22270  df-lp 22385  df-perf 22386

This theorem is referenced by:  perfdvf  25165