Theorem pnrmopn 22838

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 22836	. . . 4 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)
2		eqid 2732	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	opncld 22528	. . . 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 22837	. . 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 724	. . . . . . . 8 ⊢ (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) ∧ 𝑥 ∈ ℕ) → 𝐽 ∈ Top)
8		elmapi 8839	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝐽 ↑m ℕ) → 𝑔:ℕ⟶𝐽)
9	8	adantl 482	. . . . . . . . 9 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → 𝑔:ℕ⟶𝐽)
10	9	ffvelcdmda 7083	. . . . . . . 8 ⊢ (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ 𝐽)
11	2	opncld 22528	. . . . . . . 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 7111	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))):ℕ⟶(Clsd‘𝐽))
14		fvex 6901	. . . . . . 7 ⊢ (Clsd‘𝐽) ∈ V
15		nnex 12214	. . . . . . 7 ⊢ ℕ ∈ V
16	14, 15	elmap 8861	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ) ↔ (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))):ℕ⟶(Clsd‘𝐽))
17	13, 16	sylibr 233	. . . . 5 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ))
18		iundif2 5076	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ 𝑥 ∈ ℕ (𝑔‘𝑥))
19		ffn 6714	. . . . . . . . 9 ⊢ (𝑔:ℕ⟶𝐽 → 𝑔 Fn ℕ)
20		fniinfv 6966	. . . . . . . . 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 4121	. . . . . . 7 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (∪ 𝐽 ∖ ∩ 𝑥 ∈ ℕ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
23	18, 22	eqtrid 2784	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
24		uniexg 7726	. . . . . . . . . . 11 ⊢ (𝐽 ∈ PNrm → ∪ 𝐽 ∈ V)
25	24	difexd 5328	. . . . . . . . . 10 ⊢ (𝐽 ∈ PNrm → (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V)
26	25	ralrimivw 3150	. . . . . . . . 9 ⊢ (𝐽 ∈ PNrm → ∀𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V)
27	26	adantr 481	. . . . . . . 8 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∀𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V)
28		dfiun2g 5032	. . . . . . . 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 2732	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥)))
31	30	rnmpt 5952	. . . . . . . 8 ⊢ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))}
32	31	unieqi 4920	. . . . . . 7 ⊢ ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = ∪ {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))}
33	29, 32	eqtr4di 2790	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
34	23, 33	eqtr3d 2774	. . . . 5 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
35		rneq 5933	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
36	35	unieqd 4921	. . . . . 6 ⊢ (𝑓 = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) → ∪ ran 𝑓 = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
37	36	rspceeqv 3632	. . . . 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 745	. . 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 4115	. . . . . . . 8 ⊢ ((∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
41	40	eqcomd 2738	. . . . . . 7 ⊢ ((∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔 → (∪ 𝐽 ∖ ∩ ran 𝑔) = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)))
42		elssuni 4940	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
43		dfss4 4257	. . . . . . . 8 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = 𝐴)
44	42, 43	sylib 217	. . . . . . 7 ⊢ (𝐴 ∈ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = 𝐴)
45	41, 44	sylan9eqr 2794	. . . . . 6 ⊢ ((𝐴 ∈ 𝐽 ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔) → (∪ 𝐽 ∖ ∩ ran 𝑔) = 𝐴)
46	45	ad2ant2l 744	. . . . 5 ⊢ (((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) ∧ (𝑔 ∈ (𝐽 ↑m ℕ) ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)) → (∪ 𝐽 ∖ ∩ ran 𝑔) = 𝐴)
47	46	eqeq1d 2734	. . . 4 ⊢ (((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) ∧ (𝑔 ∈ (𝐽 ↑m ℕ) ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)) → ((∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran 𝑓 ↔ 𝐴 = ∪ ran 𝑓))
48	47	rexbidv 3178	. . 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 231	. 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 396   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ∖ cdif 3944   ⊆ wss 3947  ∪ cuni 4907  ∩ cint 4949  ∪ ciun 4996  ∩ ciin 4997   ↦ cmpt 5230  ran crn 5676   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405   ↑_m cmap 8816  ℕcn 12208  Topctop 22386  Clsdccld 22511  PNrmcpnrm 22807

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-map 8818  df-nn 12209  df-top 22387  df-cld 22514  df-nrm 22812  df-pnrm 22814

This theorem is referenced by: (None)