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Theorem pnrmopn 21368
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 21366 . . . 4 (𝐽 ∈ PNrm → 𝐽 ∈ Top)
2 eqid 2771 . . . . 5 𝐽 = 𝐽
32opncld 21058 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝐽) → ( 𝐽𝐴) ∈ (Clsd‘𝐽))
41, 3sylan 561 . . 3 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ( 𝐽𝐴) ∈ (Clsd‘𝐽))
5 pnrmcld 21367 . . 3 ((𝐽 ∈ PNrm ∧ ( 𝐽𝐴) ∈ (Clsd‘𝐽)) → ∃𝑔 ∈ (𝐽𝑚 ℕ)( 𝐽𝐴) = ran 𝑔)
64, 5syldan 571 . 2 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ∃𝑔 ∈ (𝐽𝑚 ℕ)( 𝐽𝐴) = ran 𝑔)
71ad2antrr 697 . . . . . . . 8 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) ∧ 𝑥 ∈ ℕ) → 𝐽 ∈ Top)
8 elmapi 8031 . . . . . . . . . 10 (𝑔 ∈ (𝐽𝑚 ℕ) → 𝑔:ℕ⟶𝐽)
98adantl 467 . . . . . . . . 9 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → 𝑔:ℕ⟶𝐽)
109ffvelrnda 6502 . . . . . . . 8 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ 𝐽)
112opncld 21058 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑔𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑔𝑥)) ∈ (Clsd‘𝐽))
127, 10, 11syl2anc 565 . . . . . . 7 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) ∧ 𝑥 ∈ ℕ) → ( 𝐽 ∖ (𝑔𝑥)) ∈ (Clsd‘𝐽))
13 eqid 2771 . . . . . . 7 (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥)))
1412, 13fmptd 6527 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))):ℕ⟶(Clsd‘𝐽))
15 fvex 6342 . . . . . . 7 (Clsd‘𝐽) ∈ V
16 nnex 11228 . . . . . . 7 ℕ ∈ V
1715, 16elmap 8038 . . . . . 6 ((𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) ∈ ((Clsd‘𝐽) ↑𝑚 ℕ) ↔ (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))):ℕ⟶(Clsd‘𝐽))
1814, 17sylibr 224 . . . . 5 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) ∈ ((Clsd‘𝐽) ↑𝑚 ℕ))
19 iundif2 4721 . . . . . . 7 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ( 𝐽 𝑥 ∈ ℕ (𝑔𝑥))
20 ffn 6185 . . . . . . . . 9 (𝑔:ℕ⟶𝐽𝑔 Fn ℕ)
21 fniinfv 6399 . . . . . . . . 9 (𝑔 Fn ℕ → 𝑥 ∈ ℕ (𝑔𝑥) = ran 𝑔)
229, 20, 213syl 18 . . . . . . . 8 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → 𝑥 ∈ ℕ (𝑔𝑥) = ran 𝑔)
2322difeq2d 3879 . . . . . . 7 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → ( 𝐽 𝑥 ∈ ℕ (𝑔𝑥)) = ( 𝐽 ran 𝑔))
2419, 23syl5eq 2817 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ( 𝐽 ran 𝑔))
25 uniexg 7102 . . . . . . . . . . 11 (𝐽 ∈ PNrm → 𝐽 ∈ V)
26 difexg 4942 . . . . . . . . . . 11 ( 𝐽 ∈ V → ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2725, 26syl 17 . . . . . . . . . 10 (𝐽 ∈ PNrm → ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2827ralrimivw 3116 . . . . . . . . 9 (𝐽 ∈ PNrm → ∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2928adantr 466 . . . . . . . 8 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → ∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
30 dfiun2g 4686 . . . . . . . 8 (∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))})
3129, 30syl 17 . . . . . . 7 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))})
3213rnmpt 5509 . . . . . . . 8 ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))}
3332unieqi 4583 . . . . . . 7 ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))}
3431, 33syl6eqr 2823 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3524, 34eqtr3d 2807 . . . . 5 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → ( 𝐽 ran 𝑔) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
36 rneq 5489 . . . . . . . 8 (𝑓 = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3736unieqd 4584 . . . . . . 7 (𝑓 = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3837eqeq2d 2781 . . . . . 6 (𝑓 = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) → (( 𝐽 ran 𝑔) = ran 𝑓 ↔ ( 𝐽 ran 𝑔) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥)))))
3938rspcev 3460 . . . . 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)
4018, 35, 39syl2anc 565 . . . 4 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑𝑚 ℕ)( 𝐽 ran 𝑔) = ran 𝑓)
4140ad2ant2r 733 . . 3 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽𝑚 ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑𝑚 ℕ)( 𝐽 ran 𝑔) = ran 𝑓)
42 difeq2 3873 . . . . . . . 8 (( 𝐽𝐴) = ran 𝑔 → ( 𝐽 ∖ ( 𝐽𝐴)) = ( 𝐽 ran 𝑔))
4342eqcomd 2777 . . . . . . 7 (( 𝐽𝐴) = ran 𝑔 → ( 𝐽 ran 𝑔) = ( 𝐽 ∖ ( 𝐽𝐴)))
44 elssuni 4603 . . . . . . . 8 (𝐴𝐽𝐴 𝐽)
45 dfss4 4007 . . . . . . . 8 (𝐴 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝐴)) = 𝐴)
4644, 45sylib 208 . . . . . . 7 (𝐴𝐽 → ( 𝐽 ∖ ( 𝐽𝐴)) = 𝐴)
4743, 46sylan9eqr 2827 . . . . . 6 ((𝐴𝐽 ∧ ( 𝐽𝐴) = ran 𝑔) → ( 𝐽 ran 𝑔) = 𝐴)
4847ad2ant2l 732 . . . . 5 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽𝑚 ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ( 𝐽 ran 𝑔) = 𝐴)
4948eqeq1d 2773 . . . 4 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽𝑚 ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → (( 𝐽 ran 𝑔) = ran 𝑓𝐴 = ran 𝑓))
5049rexbidv 3200 . . 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))
5141, 50mpbid 222 . 2 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽𝑚 ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑𝑚 ℕ)𝐴 = ran 𝑓)
526, 51rexlimddv 3183 1 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑𝑚 ℕ)𝐴 = ran 𝑓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  wss 3723   cuni 4574   cint 4611   ciun 4654   ciin 4655  cmpt 4863  ran crn 5250   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  𝑚 cmap 8009  cn 11222  Topctop 20918  Clsdccld 21041  PNrmcpnrm 21337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-i2m1 10206  ax-1ne0 10207  ax-rrecex 10210  ax-cnre 10211
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-map 8011  df-nn 11223  df-top 20919  df-cld 21044  df-nrm 21342  df-pnrm 21344
This theorem is referenced by: (None)
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