Theorem pnrmopn 23326

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 23324	. . . 4 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)
2		eqid 2739	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	opncld 23016	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (∪ 𝐽 ∖ 𝐴) ∈ (Clsd‘𝐽))
4	1, 3	sylan 586	. . 3 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → (∪ 𝐽 ∖ 𝐴) ∈ (Clsd‘𝐽))
5		pnrmcld 23325	. . 3 ⊢ ((𝐽 ∈ PNrm ∧ (∪ 𝐽 ∖ 𝐴) ∈ (Clsd‘𝐽)) → ∃𝑔 ∈ (𝐽 ↑m ℕ)(∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)
6	4, 5	syldan 597	. 2 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → ∃𝑔 ∈ (𝐽 ↑m ℕ)(∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)
7	1	ad2antrr 732	. . . . . . . 8 ⊢ (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) ∧ 𝑥 ∈ ℕ) → 𝐽 ∈ Top)
8		elmapi 8786	. . . . . . . . . 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 7025	. . . . . . . 8 ⊢ (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ 𝐽)
11	2	opncld 23016	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑔‘𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ (Clsd‘𝐽))
12	7, 10, 11	syl2anc 590	. . . . . . 7 ⊢ (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) ∧ 𝑥 ∈ ℕ) → (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ (Clsd‘𝐽))
13	12	fmpttd 7056	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))):ℕ⟶(Clsd‘𝐽))
14		fvex 6840	. . . . . . 7 ⊢ (Clsd‘𝐽) ∈ V
15		nnex 12171	. . . . . . 7 ⊢ ℕ ∈ V
16	14, 15	elmap 8809	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ) ↔ (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))):ℕ⟶(Clsd‘𝐽))
17	13, 16	sylibr 235	. . . . 5 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ))
18		iundif2 5003	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ 𝑥 ∈ ℕ (𝑔‘𝑥))
19		ffn 6655	. . . . . . . . 9 ⊢ (𝑔:ℕ⟶𝐽 → 𝑔 Fn ℕ)
20		fniinfv 6905	. . . . . . . . 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 4057	. . . . . . 7 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (∪ 𝐽 ∖ ∩ 𝑥 ∈ ℕ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
23	18, 22	eqtrid 2786	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
24		uniexg 7683	. . . . . . . . . . 11 ⊢ (𝐽 ∈ PNrm → ∪ 𝐽 ∈ V)
25	24	difexd 5259	. . . . . . . . . 10 ⊢ (𝐽 ∈ PNrm → (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V)
26	25	ralrimivw 3135	. . . . . . . . 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 4959	. . . . . . . 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 2739	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥)))
31	30	rnmpt 5899	. . . . . . . 8 ⊢ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))}
32	31	unieqi 4850	. . . . . . 7 ⊢ ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = ∪ {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))}
33	29, 32	eqtr4di 2792	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
34	23, 33	eqtr3d 2776	. . . . 5 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
35		rneq 5878	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
36	35	unieqd 4851	. . . . . 6 ⊢ (𝑓 = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) → ∪ ran 𝑓 = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
37	36	rspceeqv 3583	. . . . 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 590	. . . 4 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)(∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran 𝑓)
39	38	ad2ant2r 753	. . 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 4051	. . . . . . . 8 ⊢ ((∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
41	40	eqcomd 2745	. . . . . . 7 ⊢ ((∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔 → (∪ 𝐽 ∖ ∩ ran 𝑔) = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)))
42		elssuni 4869	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
43		dfss4 4197	. . . . . . . 8 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = 𝐴)
44	42, 43	sylib 219	. . . . . . 7 ⊢ (𝐴 ∈ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = 𝐴)
45	41, 44	sylan9eqr 2796	. . . . . 6 ⊢ ((𝐴 ∈ 𝐽 ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔) → (∪ 𝐽 ∖ ∩ ran 𝑔) = 𝐴)
46	45	ad2ant2l 752	. . . . 5 ⊢ (((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) ∧ (𝑔 ∈ (𝐽 ↑m ℕ) ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)) → (∪ 𝐽 ∖ ∩ ran 𝑔) = 𝐴)
47	46	eqeq1d 2741	. . . 4 ⊢ (((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) ∧ (𝑔 ∈ (𝐽 ↑m ℕ) ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)) → ((∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran 𝑓 ↔ 𝐴 = ∪ ran 𝑓))
48	47	rexbidv 3163	. . 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 233	. 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 3146	1 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ∪ ran 𝑓)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ∖ cdif 3880   ⊆ wss 3883  ∪ cuni 4838  ∩ cint 4877  ∪ ciun 4921  ∩ ciin 4922   ↦ cmpt 5153  ran crn 5619   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ↑_m cmap 8763  ℕcn 12165  Topctop 22876  Clsdccld 22999  PNrmcpnrm 23295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-1cn 11087  ax-addcl 11089

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-map 8765  df-nn 12166  df-top 22877  df-cld 23002  df-nrm 23300  df-pnrm 23302

This theorem is referenced by: (None)