Theorem pnrmopn 23716

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 23714	. . . 4 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)
2		eqid 2763	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	opncld 23406	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (∪ 𝐽 ∖ 𝐴) ∈ (Clsd‘𝐽))
4	1, 3	sylan 592	. . 3 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → (∪ 𝐽 ∖ 𝐴) ∈ (Clsd‘𝐽))
5		pnrmcld 23715	. . 3 ⊢ ((𝐽 ∈ PNrm ∧ (∪ 𝐽 ∖ 𝐴) ∈ (Clsd‘𝐽)) → ∃𝑔 ∈ (𝐽 ↑m ℕ)(∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)
6	4, 5	syldan 603	. 2 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → ∃𝑔 ∈ (𝐽 ↑m ℕ)(∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)
7	1	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) ∧ 𝑥 ∈ ℕ) → 𝐽 ∈ Top)
8		elmapi 8907	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝐽 ↑m ℕ) → 𝑔:ℕ⟶𝐽)
9	8	adantl 487	. . . . . . . . 9 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → 𝑔:ℕ⟶𝐽)
10	9	ffvelcdmda 7116	. . . . . . . 8 ⊢ (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ 𝐽)
11	2	opncld 23406	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑔‘𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ (Clsd‘𝐽))
12	7, 10, 11	syl2anc 596	. . . . . . 7 ⊢ (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) ∧ 𝑥 ∈ ℕ) → (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ (Clsd‘𝐽))
13	12	fmpttd 7147	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))):ℕ⟶(Clsd‘𝐽))
14		fvex 6929	. . . . . . 7 ⊢ (Clsd‘𝐽) ∈ V
15		nnex 12381	. . . . . . 7 ⊢ ℕ ∈ V
16	14, 15	elmap 8937	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ) ↔ (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))):ℕ⟶(Clsd‘𝐽))
17	13, 16	sylibr 237	. . . . 5 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ))
18		iundif2 5037	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ 𝑥 ∈ ℕ (𝑔‘𝑥))
19		ffn 6739	. . . . . . . . 9 ⊢ (𝑔:ℕ⟶𝐽 → 𝑔 Fn ℕ)
20		fniinfv 6994	. . . . . . . . 9 ⊢ (𝑔 Fn ℕ → ∩ 𝑥 ∈ ℕ (𝑔‘𝑥) = ∩ ran 𝑔)
21	9, 19, 20	3syl 19	. . . . . . . 8 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∩ 𝑥 ∈ ℕ (𝑔‘𝑥) = ∩ ran 𝑔)
22	21	difeq2d 4066	. . . . . . 7 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (∪ 𝐽 ∖ ∩ 𝑥 ∈ ℕ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
23	18, 22	eqtrid 2810	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
24		uniexg 7789	. . . . . . . . . . 11 ⊢ (𝐽 ∈ PNrm → ∪ 𝐽 ∈ V)
25	24	difexd 5300	. . . . . . . . . 10 ⊢ (𝐽 ∈ PNrm → (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V)
26	25	ralrimivw 3162	. . . . . . . . 9 ⊢ (𝐽 ∈ PNrm → ∀𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V)
27	26	adantr 486	. . . . . . . 8 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∀𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V)
28		dfiun2g 4993	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) ∈ V → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = ∪ {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))})
29	27, 28	syl 18	. . . . . . 7 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = ∪ {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))})
30		eqid 2763	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥)))
31	30	rnmpt 5951	. . . . . . . 8 ⊢ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))}
32	31	unieqi 4884	. . . . . . 7 ⊢ ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) = ∪ {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = (∪ 𝐽 ∖ (𝑔‘𝑥))}
33	29, 32	eqtr4di 2816	. . . . . 6 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∪ 𝑥 ∈ ℕ (∪ 𝐽 ∖ (𝑔‘𝑥)) = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
34	23, 33	eqtr3d 2800	. . . . 5 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → (∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
35		rneq 5930	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
36	35	unieqd 4885	. . . . . 6 ⊢ (𝑓 = (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))) → ∪ ran 𝑓 = ∪ ran (𝑥 ∈ ℕ ↦ (∪ 𝐽 ∖ (𝑔‘𝑥))))
37	36	rspceeqv 3602	. . . . 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 596	. . . 4 ⊢ ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽 ↑m ℕ)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)(∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran 𝑓)
39	38	ad2ant2r 760	. . 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 4060	. . . . . . . 8 ⊢ ((∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = (∪ 𝐽 ∖ ∩ ran 𝑔))
41	40	eqcomd 2769	. . . . . . 7 ⊢ ((∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔 → (∪ 𝐽 ∖ ∩ ran 𝑔) = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)))
42		elssuni 4904	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
43		dfss4 4217	. . . . . . . 8 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = 𝐴)
44	42, 43	sylib 221	. . . . . . 7 ⊢ (𝐴 ∈ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐴)) = 𝐴)
45	41, 44	sylan9eqr 2820	. . . . . 6 ⊢ ((𝐴 ∈ 𝐽 ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔) → (∪ 𝐽 ∖ ∩ ran 𝑔) = 𝐴)
46	45	ad2ant2l 759	. . . . 5 ⊢ (((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) ∧ (𝑔 ∈ (𝐽 ↑m ℕ) ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)) → (∪ 𝐽 ∖ ∩ ran 𝑔) = 𝐴)
47	46	eqeq1d 2765	. . . 4 ⊢ (((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) ∧ (𝑔 ∈ (𝐽 ↑m ℕ) ∧ (∪ 𝐽 ∖ 𝐴) = ∩ ran 𝑔)) → ((∪ 𝐽 ∖ ∩ ran 𝑔) = ∪ ran 𝑓 ↔ 𝐴 = ∪ ran 𝑓))
48	47	rexbidv 3190	. . 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 235	. 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 3173	1 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ∪ ran 𝑓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  ∪ cuni 4872  ∩ cint 4912  ∪ ciun 4956  ∩ ciin 4957   ↦ cmpt 5192  ran crn 5659   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885  ℕcn 12375  Topctop 23266  Clsdccld 23389  PNrmcpnrm 23685

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-1cn 11297  ax-addcl 11299

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-map 8887  df-nn 12376  df-top 23267  df-cld 23392  df-nrm 23690  df-pnrm 23692

This theorem is used by: (None)