Theorem pnrmopn 23351

Description: An open set in a perfectly normal space is a countable union of closed sets. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
pnrmopn	⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ∪ ran 𝑓)

Distinct variable groups:   𝐴,𝑓   𝑓,𝐽

Proof of Theorem pnrmopn

Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pnrmtop 23349	. . . 4 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)
2		eqid 2737	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	opncld 23041	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (∪ 𝐽 ∖ 𝐴) ∈ (Clsd‘𝐽))
4	1, 3	sylan 580	. . 3 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → (∪ 𝐽 ∖ 𝐴) ∈ (Clsd‘𝐽))
5		pnrmcld 23350	. . 3 ⊢ ((𝐽 ∈ PNrm ∧ (∪ 𝐽 ∖ 𝐴) ∈ (Clsd‘𝐽)) → ∃𝑔 ∈ (𝐽 ↑m ℕ)(∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)
6	4, 5	syldan 591	. 2 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → ∃𝑔 ∈ (𝐽 ↑m ℕ)(∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)
7	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) ∧ 𝑥 ∈ ℕ) → 𝐽 ∈ Top)
8		elmapi 8889	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝐽 ↑m ℕ) → 𝑔:ℕ⟶𝐽)
9	8	adantl 481	. . . . . . . . 9 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → 𝑔:ℕ⟶𝐽)
10	9	ffvelcdmda 7104	. . . . . . . 8 ⊢ (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ 𝐽)
11	2	opncld 23041	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑔‘𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ (Clsd‘𝐽))
12	7, 10, 11	syl2anc 584	. . . . . . 7 ⊢ (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) ∧ 𝑥 ∈ ℕ) → (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ (Clsd‘𝐽))
13	12	fmpttd 7135	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))):ℕ⟶(Clsd‘𝐽))
14		fvex 6919	. . . . . . 7 ⊢ (Clsd‘𝐽) ∈ V
15		nnex 12272	. . . . . . 7 ⊢ ℕ ∈ V
16	14, 15	elmap 8911	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ) ↔ (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))):ℕ⟶(Clsd‘𝐽))
17	13, 16	sylibr 234	. . . . 5 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ))
18		iundif2 5074	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ 𝑥 ∈ ℕ (𝑔‘𝑥))
19		ffn 6736	. . . . . . . . 9 ⊢ (𝑔:ℕ⟶𝐽 → 𝑔 Fn ℕ)
20		fniinfv 6987	. . . . . . . . 9 ⊢ (𝑔 Fn ℕ → ∩ 𝑥 ∈ ℕ (𝑔‘𝑥) = ∩ ran 𝑔)
21	9, 19, 20	3syl 18	. . . . . . . 8 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∩ 𝑥 ∈ ℕ (𝑔‘𝑥) = ∩ ran 𝑔)
22	21	difeq2d 4126	. . . . . . 7 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (∪ 𝐽 ∖ ∩ 𝑥 ∈ ℕ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
23	18, 22	eqtrid 2789	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
24		uniexg 7760	. . . . . . . . . . 11 ⊢ (𝐽 ∈ PNrm → ∪ 𝐽 ∈ V)
25	24	difexd 5331	. . . . . . . . . 10 ⊢ (𝐽 ∈ PNrm → (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V)
26	25	ralrimivw 3150	. . . . . . . . 9 ⊢ (𝐽 ∈ PNrm → ∀𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V)
27	26	adantr 480	. . . . . . . 8 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∀𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V)
28		dfiun2g 5030	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = ∪ {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))})
29	27, 28	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = ∪ {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))})
30		eqid 2737	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥)))
31	30	rnmpt 5968	. . . . . . . 8 ⊢ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))}
32	31	unieqi 4919	. . . . . . 7 ⊢ ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = ∪ {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))}
33	29, 32	eqtr4di 2795	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
34	23, 33	eqtr3d 2779	. . . . 5 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
35		rneq 5947	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
36	35	unieqd 4920	. . . . . 6 ⊢ (𝑓 = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) → ∪ ran 𝑓 = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
37	36	rspceeqv 3645	. . . . 5 ⊢ (((𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ) ∧ (∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥)))) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)(∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran 𝑓)
38	17, 34, 37	syl2anc 584	. . . 4 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)(∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran 𝑓)
39	38	ad2ant2r 747	. . 3 ⊢ (((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) ∧ (𝑔 ∈ (𝐽 ↑m ℕ) ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)(∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran 𝑓)
40		difeq2 4120	. . . . . . . 8 ⊢ ((∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
41	40	eqcomd 2743	. . . . . . 7 ⊢ ((∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔 → (∪ 𝐽 ∖ ∩ ran 𝑔) = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)))
42		elssuni 4937	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
43		dfss4 4269	. . . . . . . 8 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = 𝐴)
44	42, 43	sylib 218	. . . . . . 7 ⊢ (𝐴 ∈ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = 𝐴)
45	41, 44	sylan9eqr 2799	. . . . . 6 ⊢ ((𝐴 ∈ 𝐽 ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔) → (∪ 𝐽 ∖ ∩ ran 𝑔) = 𝐴)
46	45	ad2ant2l 746	. . . . 5 ⊢ (((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) ∧ (𝑔 ∈ (𝐽 ↑m ℕ) ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)) → (∪ 𝐽 ∖ ∩ ran 𝑔) = 𝐴)
47	46	eqeq1d 2739	. . . 4 ⊢ (((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) ∧ (𝑔 ∈ (𝐽 ↑m ℕ) ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)) → ((∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran 𝑓 ↔ 𝐴 = ∪ ran 𝑓))
48	47	rexbidv 3179	. . 3 ⊢ (((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) ∧ (𝑔 ∈ (𝐽 ↑m ℕ) ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)) → (∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)(∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran 𝑓 ↔ ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ∪ ran 𝑓))
49	39, 48	mpbid 232	. 2 ⊢ (((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) ∧ (𝑔 ∈ (𝐽 ↑m ℕ) ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ∪ ran 𝑓)
50	6, 49	rexlimddv 3161	1 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ∪ ran 𝑓)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  ∪ cuni 4907  ∩ cint 4946  ∪ ciun 4991  ∩ ciin 4992   ↦ cmpt 5225  ran crn 5686   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ↑_m cmap 8866  ℕcn 12266  Topctop 22899  Clsdccld 23024  PNrmcpnrm 23320

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-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215

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-pred 6321  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-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-map 8868  df-nn 12267  df-top 22900  df-cld 23027  df-nrm 23325  df-pnrm 23327

This theorem is referenced by: (None)