Theorem pnrmopn 23259

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 23257	. . . 4 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)
2		eqid 2731	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	opncld 22949	. . . 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 23258	. . 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 8773	. . . . . . . . . 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 7017	. . . . . . . 8 ⊢ (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ 𝐽)
11	2	opncld 22949	. . . . . . . 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 7048	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))):ℕ⟶(Clsd‘𝐽))
14		fvex 6835	. . . . . . 7 ⊢ (Clsd‘𝐽) ∈ V
15		nnex 12131	. . . . . . 7 ⊢ ℕ ∈ V
16	14, 15	elmap 8795	. . . . . 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 5022	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ 𝑥 ∈ ℕ (𝑔‘𝑥))
19		ffn 6651	. . . . . . . . 9 ⊢ (𝑔:ℕ⟶𝐽 → 𝑔 Fn ℕ)
20		fniinfv 6900	. . . . . . . . 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 4076	. . . . . . 7 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (∪ 𝐽 ∖ ∩ 𝑥 ∈ ℕ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
23	18, 22	eqtrid 2778	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
24		uniexg 7673	. . . . . . . . . . 11 ⊢ (𝐽 ∈ PNrm → ∪ 𝐽 ∈ V)
25	24	difexd 5269	. . . . . . . . . 10 ⊢ (𝐽 ∈ PNrm → (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V)
26	25	ralrimivw 3128	. . . . . . . . 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 4980	. . . . . . . 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 2731	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥)))
31	30	rnmpt 5897	. . . . . . . 8 ⊢ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))}
32	31	unieqi 4871	. . . . . . 7 ⊢ ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = ∪ {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))}
33	29, 32	eqtr4di 2784	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
34	23, 33	eqtr3d 2768	. . . . 5 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
35		rneq 5876	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
36	35	unieqd 4872	. . . . . 6 ⊢ (𝑓 = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) → ∪ ran 𝑓 = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
37	36	rspceeqv 3600	. . . . 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 4070	. . . . . . . 8 ⊢ ((∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
41	40	eqcomd 2737	. . . . . . 7 ⊢ ((∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔 → (∪ 𝐽 ∖ ∩ ran 𝑔) = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)))
42		elssuni 4889	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
43		dfss4 4219	. . . . . . . 8 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = 𝐴)
44	42, 43	sylib 218	. . . . . . 7 ⊢ (𝐴 ∈ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = 𝐴)
45	41, 44	sylan9eqr 2788	. . . . . 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 2733	. . . 4 ⊢ (((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) ∧ (𝑔 ∈ (𝐽 ↑m ℕ) ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)) → ((∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran 𝑓 ↔ 𝐴 = ∪ ran 𝑓))
48	47	rexbidv 3156	. . 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 3139	1 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ∪ ran 𝑓)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∖ cdif 3899   ⊆ wss 3902  ∪ cuni 4859  ∩ cint 4897  ∪ ciun 4941  ∩ ciin 4942   ↦ cmpt 5172  ran crn 5617   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750  ℕcn 12125  Topctop 22809  Clsdccld 22932  PNrmcpnrm 23228

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-1cn 11064  ax-addcl 11066

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-map 8752  df-nn 12126  df-top 22810  df-cld 22935  df-nrm 23233  df-pnrm 23235

This theorem is referenced by: (None)