MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pnrmopn Structured version   Visualization version   GIF version

Theorem pnrmopn 21427
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 21425 . . . 4 (𝐽 ∈ PNrm → 𝐽 ∈ Top)
2 eqid 2765 . . . . 5 𝐽 = 𝐽
32opncld 21117 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝐽) → ( 𝐽𝐴) ∈ (Clsd‘𝐽))
41, 3sylan 575 . . 3 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ( 𝐽𝐴) ∈ (Clsd‘𝐽))
5 pnrmcld 21426 . . 3 ((𝐽 ∈ PNrm ∧ ( 𝐽𝐴) ∈ (Clsd‘𝐽)) → ∃𝑔 ∈ (𝐽𝑚 ℕ)( 𝐽𝐴) = ran 𝑔)
64, 5syldan 585 . 2 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ∃𝑔 ∈ (𝐽𝑚 ℕ)( 𝐽𝐴) = ran 𝑔)
71ad2antrr 717 . . . . . . . 8 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) ∧ 𝑥 ∈ ℕ) → 𝐽 ∈ Top)
8 elmapi 8082 . . . . . . . . . 10 (𝑔 ∈ (𝐽𝑚 ℕ) → 𝑔:ℕ⟶𝐽)
98adantl 473 . . . . . . . . 9 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → 𝑔:ℕ⟶𝐽)
109ffvelrnda 6549 . . . . . . . 8 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ 𝐽)
112opncld 21117 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑔𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑔𝑥)) ∈ (Clsd‘𝐽))
127, 10, 11syl2anc 579 . . . . . . 7 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) ∧ 𝑥 ∈ ℕ) → ( 𝐽 ∖ (𝑔𝑥)) ∈ (Clsd‘𝐽))
1312fmpttd 6575 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))):ℕ⟶(Clsd‘𝐽))
14 fvex 6388 . . . . . . 7 (Clsd‘𝐽) ∈ V
15 nnex 11281 . . . . . . 7 ℕ ∈ V
1614, 15elmap 8089 . . . . . 6 ((𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) ∈ ((Clsd‘𝐽) ↑𝑚 ℕ) ↔ (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))):ℕ⟶(Clsd‘𝐽))
1713, 16sylibr 225 . . . . 5 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) ∈ ((Clsd‘𝐽) ↑𝑚 ℕ))
18 iundif2 4743 . . . . . . 7 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ( 𝐽 𝑥 ∈ ℕ (𝑔𝑥))
19 ffn 6223 . . . . . . . . 9 (𝑔:ℕ⟶𝐽𝑔 Fn ℕ)
20 fniinfv 6446 . . . . . . . . 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 3890 . . . . . . 7 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → ( 𝐽 𝑥 ∈ ℕ (𝑔𝑥)) = ( 𝐽 ran 𝑔))
2318, 22syl5eq 2811 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ( 𝐽 ran 𝑔))
24 uniexg 7153 . . . . . . . . . . 11 (𝐽 ∈ PNrm → 𝐽 ∈ V)
25 difexg 4969 . . . . . . . . . . 11 ( 𝐽 ∈ V → ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2624, 25syl 17 . . . . . . . . . 10 (𝐽 ∈ PNrm → ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2726ralrimivw 3114 . . . . . . . . 9 (𝐽 ∈ PNrm → ∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2827adantr 472 . . . . . . . 8 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → ∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
29 dfiun2g 4708 . . . . . . . 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 2765 . . . . . . . . 9 (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥)))
3231rnmpt 5540 . . . . . . . 8 ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))}
3332unieqi 4603 . . . . . . 7 ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))}
3430, 33syl6eqr 2817 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3523, 34eqtr3d 2801 . . . . 5 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → ( 𝐽 ran 𝑔) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
36 rneq 5519 . . . . . . 7 (𝑓 = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3736unieqd 4604 . . . . . 6 (𝑓 = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3837rspceeqv 3479 . . . . 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 579 . . . 4 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑𝑚 ℕ)( 𝐽 ran 𝑔) = ran 𝑓)
4039ad2ant2r 753 . . 3 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽𝑚 ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑𝑚 ℕ)( 𝐽 ran 𝑔) = ran 𝑓)
41 difeq2 3884 . . . . . . . 8 (( 𝐽𝐴) = ran 𝑔 → ( 𝐽 ∖ ( 𝐽𝐴)) = ( 𝐽 ran 𝑔))
4241eqcomd 2771 . . . . . . 7 (( 𝐽𝐴) = ran 𝑔 → ( 𝐽 ran 𝑔) = ( 𝐽 ∖ ( 𝐽𝐴)))
43 elssuni 4625 . . . . . . . 8 (𝐴𝐽𝐴 𝐽)
44 dfss4 4023 . . . . . . . 8 (𝐴 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝐴)) = 𝐴)
4543, 44sylib 209 . . . . . . 7 (𝐴𝐽 → ( 𝐽 ∖ ( 𝐽𝐴)) = 𝐴)
4642, 45sylan9eqr 2821 . . . . . 6 ((𝐴𝐽 ∧ ( 𝐽𝐴) = ran 𝑔) → ( 𝐽 ran 𝑔) = 𝐴)
4746ad2ant2l 752 . . . . 5 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽𝑚 ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ( 𝐽 ran 𝑔) = 𝐴)
4847eqeq1d 2767 . . . 4 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽𝑚 ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → (( 𝐽 ran 𝑔) = ran 𝑓𝐴 = ran 𝑓))
4948rexbidv 3199 . . 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 223 . 2 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽𝑚 ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑𝑚 ℕ)𝐴 = ran 𝑓)
516, 50rexlimddv 3182 1 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑𝑚 ℕ)𝐴 = ran 𝑓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  Vcvv 3350  cdif 3729  wss 3732   cuni 4594   cint 4633   ciun 4676   ciin 4677  cmpt 4888  ran crn 5278   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  cn 11274  Topctop 20977  Clsdccld 21100  PNrmcpnrm 21396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-1cn 10247  ax-addcl 10249
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-map 8062  df-nn 11275  df-top 20978  df-cld 21103  df-nrm 21401  df-pnrm 21403
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator