Theorem perfopn 23141

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 23112	. . . . . . 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 22873	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
6	3, 5	sylib 218	. . . . 5 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
7		elssuni 4896	. . . . . . 7 ⊢ (𝑌 ∈ 𝐽 → 𝑌 ⊆ ∪ 𝐽)
8	7	adantl 481	. . . . . 6 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝑌 ⊆ ∪ 𝐽)
9	8, 4	sseqtrrdi 3977	. . . . 5 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝑌 ⊆ 𝑋)
10		resttopon 23117	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
11	6, 9, 10	syl2anc 585	. . . 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 22869	. . 3 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
14	12, 13	syl 17	. 2 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ Top)
15	9	sselda 3935	. . . . . 6 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
16	4	perfi 23111	. . . . . . 7 ⊢ ((𝐽 ∈ Perf ∧ 𝑥 ∈ 𝑋) → ¬ {𝑥} ∈ 𝐽)
17	16	adantlr 716	. . . . . 6 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑋) → ¬ {𝑥} ∈ 𝐽)
18	15, 17	syldan 592	. . . . 5 ⊢ (((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) ∧ 𝑥 ∈ 𝑌) → ¬ {𝑥} ∈ 𝐽)
19	1	eleq2i 2829	. . . . . 6 ⊢ ({𝑥} ∈ 𝐾 ↔ {𝑥} ∈ (𝐽 ↾t 𝑌))
20		restopn2 23133	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ 𝐽) → ({𝑥} ∈ (𝐽 ↾t 𝑌) ↔ ({𝑥} ∈ 𝐽 ∧ {𝑥} ⊆ 𝑌)))
21	2, 20	sylan 581	. . . . . . . 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 3130	. . 3 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → ∀𝑥 ∈ 𝑌 ¬ {𝑥} ∈ 𝐾)
28		toponuni 22870	. . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
29	12, 28	syl 17	. . 3 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝑌 = ∪ 𝐾)
30	27, 29	raleqtrdv 3300	. 2 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → ∀𝑥 ∈ ∪ 𝐾 ¬ {𝑥} ∈ 𝐾)
31		eqid 2737	. . 3 ⊢ ∪ 𝐾 = ∪ 𝐾
32	31	isperf3 23109	. 2 ⊢ (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐾 ¬ {𝑥} ∈ 𝐾))
33	14, 30, 32	sylanbrc 584	1 ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ Perf)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903  {csn 4582  ∪ cuni 4865  ‘cfv 6500  (class class class)co 7368   ↾_t crest 17352  Topctop 22849  TopOnctopon 22866  Perfcperf 23091

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-en 8896  df-fin 8899  df-fi 9326  df-rest 17354  df-topgen 17375  df-top 22850  df-topon 22867  df-bases 22902  df-cld 22975  df-ntr 22976  df-cls 22977  df-lp 23092  df-perf 23093

This theorem is referenced by:  perfdvf  25872