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Theorem pnrmopn 21879
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 ∧ 𝐴𝐽) → ∃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.
StepHypRef Expression
1 pnrmtop 21877 . . . 4 (𝐽 ∈ PNrm → 𝐽 ∈ Top)
2 eqid 2818 . . . . 5 𝐽 = 𝐽
32opncld 21569 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝐽) → ( 𝐽𝐴) ∈ (Clsd‘𝐽))
41, 3sylan 580 . . 3 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ( 𝐽𝐴) ∈ (Clsd‘𝐽))
5 pnrmcld 21878 . . 3 ((𝐽 ∈ PNrm ∧ ( 𝐽𝐴) ∈ (Clsd‘𝐽)) → ∃𝑔 ∈ (𝐽m ℕ)( 𝐽𝐴) = ran 𝑔)
64, 5syldan 591 . 2 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ∃𝑔 ∈ (𝐽m ℕ)( 𝐽𝐴) = ran 𝑔)
71ad2antrr 722 . . . . . . . 8 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) ∧ 𝑥 ∈ ℕ) → 𝐽 ∈ Top)
8 elmapi 8417 . . . . . . . . . 10 (𝑔 ∈ (𝐽m ℕ) → 𝑔:ℕ⟶𝐽)
98adantl 482 . . . . . . . . 9 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → 𝑔:ℕ⟶𝐽)
109ffvelrnda 6843 . . . . . . . 8 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ 𝐽)
112opncld 21569 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑔𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑔𝑥)) ∈ (Clsd‘𝐽))
127, 10, 11syl2anc 584 . . . . . . 7 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) ∧ 𝑥 ∈ ℕ) → ( 𝐽 ∖ (𝑔𝑥)) ∈ (Clsd‘𝐽))
1312fmpttd 6871 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))):ℕ⟶(Clsd‘𝐽))
14 fvex 6676 . . . . . . 7 (Clsd‘𝐽) ∈ V
15 nnex 11632 . . . . . . 7 ℕ ∈ V
1614, 15elmap 8424 . . . . . 6 ((𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ) ↔ (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))):ℕ⟶(Clsd‘𝐽))
1713, 16sylibr 235 . . . . 5 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ))
18 iundif2 4987 . . . . . . 7 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ( 𝐽 𝑥 ∈ ℕ (𝑔𝑥))
19 ffn 6507 . . . . . . . . 9 (𝑔:ℕ⟶𝐽𝑔 Fn ℕ)
20 fniinfv 6735 . . . . . . . . 9 (𝑔 Fn ℕ → 𝑥 ∈ ℕ (𝑔𝑥) = ran 𝑔)
219, 19, 203syl 18 . . . . . . . 8 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → 𝑥 ∈ ℕ (𝑔𝑥) = ran 𝑔)
2221difeq2d 4096 . . . . . . 7 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → ( 𝐽 𝑥 ∈ ℕ (𝑔𝑥)) = ( 𝐽 ran 𝑔))
2318, 22syl5eq 2865 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ( 𝐽 ran 𝑔))
24 uniexg 7456 . . . . . . . . . . 11 (𝐽 ∈ PNrm → 𝐽 ∈ V)
25 difexg 5222 . . . . . . . . . . 11 ( 𝐽 ∈ V → ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2624, 25syl 17 . . . . . . . . . 10 (𝐽 ∈ PNrm → ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2726ralrimivw 3180 . . . . . . . . 9 (𝐽 ∈ PNrm → ∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2827adantr 481 . . . . . . . 8 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → ∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
29 dfiun2g 4946 . . . . . . . 8 (∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))})
3028, 29syl 17 . . . . . . 7 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))})
31 eqid 2818 . . . . . . . . 9 (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥)))
3231rnmpt 5820 . . . . . . . 8 ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))}
3332unieqi 4839 . . . . . . 7 ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))}
3430, 33syl6eqr 2871 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3523, 34eqtr3d 2855 . . . . 5 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → ( 𝐽 ran 𝑔) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
36 rneq 5799 . . . . . . 7 (𝑓 = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3736unieqd 4840 . . . . . 6 (𝑓 = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3837rspceeqv 3635 . . . . 5 (((𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ) ∧ ( 𝐽 ran 𝑔) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥)))) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)( 𝐽 ran 𝑔) = ran 𝑓)
3917, 35, 38syl2anc 584 . . . 4 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)( 𝐽 ran 𝑔) = ran 𝑓)
4039ad2ant2r 743 . . 3 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽m ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)( 𝐽 ran 𝑔) = ran 𝑓)
41 difeq2 4090 . . . . . . . 8 (( 𝐽𝐴) = ran 𝑔 → ( 𝐽 ∖ ( 𝐽𝐴)) = ( 𝐽 ran 𝑔))
4241eqcomd 2824 . . . . . . 7 (( 𝐽𝐴) = ran 𝑔 → ( 𝐽 ran 𝑔) = ( 𝐽 ∖ ( 𝐽𝐴)))
43 elssuni 4859 . . . . . . . 8 (𝐴𝐽𝐴 𝐽)
44 dfss4 4232 . . . . . . . 8 (𝐴 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝐴)) = 𝐴)
4543, 44sylib 219 . . . . . . 7 (𝐴𝐽 → ( 𝐽 ∖ ( 𝐽𝐴)) = 𝐴)
4642, 45sylan9eqr 2875 . . . . . 6 ((𝐴𝐽 ∧ ( 𝐽𝐴) = ran 𝑔) → ( 𝐽 ran 𝑔) = 𝐴)
4746ad2ant2l 742 . . . . 5 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽m ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ( 𝐽 ran 𝑔) = 𝐴)
4847eqeq1d 2820 . . . 4 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽m ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → (( 𝐽 ran 𝑔) = ran 𝑓𝐴 = ran 𝑓))
4948rexbidv 3294 . . 3 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽m ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → (∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)( 𝐽 ran 𝑔) = ran 𝑓 ↔ ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ran 𝑓))
5040, 49mpbid 233 . 2 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽m ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ran 𝑓)
516, 50rexlimddv 3288 1 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ran 𝑓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {cab 2796  wral 3135  wrex 3136  Vcvv 3492  cdif 3930  wss 3933   cuni 4830   cint 4867   ciun 4910   ciin 4911  cmpt 5137  ran crn 5549   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  cn 11626  Topctop 21429  Clsdccld 21552  PNrmcpnrm 21848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-1cn 10583  ax-addcl 10585
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-map 8397  df-nn 11627  df-top 21430  df-cld 21555  df-nrm 21853  df-pnrm 21855
This theorem is referenced by: (None)
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