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Theorem pnrmopn 21066
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 21064 . . . 4 (𝐽 ∈ PNrm → 𝐽 ∈ Top)
2 eqid 2621 . . . . 5 𝐽 = 𝐽
32opncld 20756 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝐽) → ( 𝐽𝐴) ∈ (Clsd‘𝐽))
41, 3sylan 488 . . 3 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ( 𝐽𝐴) ∈ (Clsd‘𝐽))
5 pnrmcld 21065 . . 3 ((𝐽 ∈ PNrm ∧ ( 𝐽𝐴) ∈ (Clsd‘𝐽)) → ∃𝑔 ∈ (𝐽𝑚 ℕ)( 𝐽𝐴) = ran 𝑔)
64, 5syldan 487 . 2 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ∃𝑔 ∈ (𝐽𝑚 ℕ)( 𝐽𝐴) = ran 𝑔)
71ad2antrr 761 . . . . . . . 8 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) ∧ 𝑥 ∈ ℕ) → 𝐽 ∈ Top)
8 elmapi 7830 . . . . . . . . . 10 (𝑔 ∈ (𝐽𝑚 ℕ) → 𝑔:ℕ⟶𝐽)
98adantl 482 . . . . . . . . 9 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → 𝑔:ℕ⟶𝐽)
109ffvelrnda 6320 . . . . . . . 8 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ 𝐽)
112opncld 20756 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑔𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑔𝑥)) ∈ (Clsd‘𝐽))
127, 10, 11syl2anc 692 . . . . . . 7 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) ∧ 𝑥 ∈ ℕ) → ( 𝐽 ∖ (𝑔𝑥)) ∈ (Clsd‘𝐽))
13 eqid 2621 . . . . . . 7 (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥)))
1412, 13fmptd 6346 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))):ℕ⟶(Clsd‘𝐽))
15 fvex 6163 . . . . . . 7 (Clsd‘𝐽) ∈ V
16 nnex 10977 . . . . . . 7 ℕ ∈ V
1715, 16elmap 7837 . . . . . 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 4558 . . . . . . 7 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ( 𝐽 𝑥 ∈ ℕ (𝑔𝑥))
20 ffn 6007 . . . . . . . . 9 (𝑔:ℕ⟶𝐽𝑔 Fn ℕ)
21 fniinfv 6219 . . . . . . . . 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 3711 . . . . . . 7 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → ( 𝐽 𝑥 ∈ ℕ (𝑔𝑥)) = ( 𝐽 ran 𝑔))
2419, 23syl5eq 2667 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ( 𝐽 ran 𝑔))
25 uniexg 6915 . . . . . . . . . . 11 (𝐽 ∈ PNrm → 𝐽 ∈ V)
26 difexg 4773 . . . . . . . . . . 11 ( 𝐽 ∈ V → ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2725, 26syl 17 . . . . . . . . . 10 (𝐽 ∈ PNrm → ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2827ralrimivw 2962 . . . . . . . . 9 (𝐽 ∈ PNrm → ∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2928adantr 481 . . . . . . . 8 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → ∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
30 dfiun2g 4523 . . . . . . . 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 5336 . . . . . . . 8 ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))}
3332unieqi 4416 . . . . . . 7 ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))}
3431, 33syl6eqr 2673 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3524, 34eqtr3d 2657 . . . . 5 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → ( 𝐽 ran 𝑔) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
36 rneq 5316 . . . . . . . 8 (𝑓 = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3736unieqd 4417 . . . . . . 7 (𝑓 = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3837eqeq2d 2631 . . . . . 6 (𝑓 = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) → (( 𝐽 ran 𝑔) = ran 𝑓 ↔ ( 𝐽 ran 𝑔) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥)))))
3938rspcev 3298 . . . . 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 692 . . . 4 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽𝑚 ℕ)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑𝑚 ℕ)( 𝐽 ran 𝑔) = ran 𝑓)
4140ad2ant2r 782 . . 3 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽𝑚 ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑𝑚 ℕ)( 𝐽 ran 𝑔) = ran 𝑓)
42 difeq2 3705 . . . . . . . 8 (( 𝐽𝐴) = ran 𝑔 → ( 𝐽 ∖ ( 𝐽𝐴)) = ( 𝐽 ran 𝑔))
4342eqcomd 2627 . . . . . . 7 (( 𝐽𝐴) = ran 𝑔 → ( 𝐽 ran 𝑔) = ( 𝐽 ∖ ( 𝐽𝐴)))
44 elssuni 4438 . . . . . . . 8 (𝐴𝐽𝐴 𝐽)
45 dfss4 3841 . . . . . . . 8 (𝐴 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝐴)) = 𝐴)
4644, 45sylib 208 . . . . . . 7 (𝐴𝐽 → ( 𝐽 ∖ ( 𝐽𝐴)) = 𝐴)
4743, 46sylan9eqr 2677 . . . . . 6 ((𝐴𝐽 ∧ ( 𝐽𝐴) = ran 𝑔) → ( 𝐽 ran 𝑔) = 𝐴)
4847ad2ant2l 781 . . . . 5 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽𝑚 ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ( 𝐽 ran 𝑔) = 𝐴)
4948eqeq1d 2623 . . . 4 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽𝑚 ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → (( 𝐽 ran 𝑔) = ran 𝑓𝐴 = ran 𝑓))
5049rexbidv 3046 . . 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 3029 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 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3189  cdif 3556  wss 3559   cuni 4407   cint 4445   ciun 4490   ciin 4491  cmpt 4678  ran crn 5080   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  cn 10971  Topctop 20626  Clsdccld 20739  PNrmcpnrm 21035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-i2m1 9955  ax-1ne0 9956  ax-rrecex 9959  ax-cnre 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-map 7811  df-nn 10972  df-top 20627  df-cld 20742  df-nrm 21040  df-pnrm 21042
This theorem is referenced by: (None)
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