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Theorem pnrmopn 22094
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 22092 . . . 4 (𝐽 ∈ PNrm → 𝐽 ∈ Top)
2 eqid 2738 . . . . 5 𝐽 = 𝐽
32opncld 21784 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝐽) → ( 𝐽𝐴) ∈ (Clsd‘𝐽))
41, 3sylan 583 . . 3 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ( 𝐽𝐴) ∈ (Clsd‘𝐽))
5 pnrmcld 22093 . . 3 ((𝐽 ∈ PNrm ∧ ( 𝐽𝐴) ∈ (Clsd‘𝐽)) → ∃𝑔 ∈ (𝐽m ℕ)( 𝐽𝐴) = ran 𝑔)
64, 5syldan 594 . 2 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ∃𝑔 ∈ (𝐽m ℕ)( 𝐽𝐴) = ran 𝑔)
71ad2antrr 726 . . . . . . . 8 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) ∧ 𝑥 ∈ ℕ) → 𝐽 ∈ Top)
8 elmapi 8459 . . . . . . . . . 10 (𝑔 ∈ (𝐽m ℕ) → 𝑔:ℕ⟶𝐽)
98adantl 485 . . . . . . . . 9 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → 𝑔:ℕ⟶𝐽)
109ffvelrnda 6861 . . . . . . . 8 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ 𝐽)
112opncld 21784 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑔𝑥) ∈ 𝐽) → ( 𝐽 ∖ (𝑔𝑥)) ∈ (Clsd‘𝐽))
127, 10, 11syl2anc 587 . . . . . . 7 (((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) ∧ 𝑥 ∈ ℕ) → ( 𝐽 ∖ (𝑔𝑥)) ∈ (Clsd‘𝐽))
1312fmpttd 6889 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))):ℕ⟶(Clsd‘𝐽))
14 fvex 6687 . . . . . . 7 (Clsd‘𝐽) ∈ V
15 nnex 11722 . . . . . . 7 ℕ ∈ V
1614, 15elmap 8481 . . . . . 6 ((𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ) ↔ (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))):ℕ⟶(Clsd‘𝐽))
1713, 16sylibr 237 . . . . 5 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) ∈ ((Clsd‘𝐽) ↑m ℕ))
18 iundif2 4959 . . . . . . 7 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ( 𝐽 𝑥 ∈ ℕ (𝑔𝑥))
19 ffn 6504 . . . . . . . . 9 (𝑔:ℕ⟶𝐽𝑔 Fn ℕ)
20 fniinfv 6746 . . . . . . . . 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 4013 . . . . . . 7 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → ( 𝐽 𝑥 ∈ ℕ (𝑔𝑥)) = ( 𝐽 ran 𝑔))
2318, 22syl5eq 2785 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ( 𝐽 ran 𝑔))
24 uniexg 7484 . . . . . . . . . . 11 (𝐽 ∈ PNrm → 𝐽 ∈ V)
2524difexd 5197 . . . . . . . . . 10 (𝐽 ∈ PNrm → ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2625ralrimivw 3097 . . . . . . . . 9 (𝐽 ∈ PNrm → ∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
2726adantr 484 . . . . . . . 8 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → ∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V)
28 dfiun2g 4917 . . . . . . . 8 (∀𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) ∈ V → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))})
2927, 28syl 17 . . . . . . 7 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))})
30 eqid 2738 . . . . . . . . 9 (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥)))
3130rnmpt 5798 . . . . . . . 8 ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))}
3231unieqi 4809 . . . . . . 7 ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) = {𝑓 ∣ ∃𝑥 ∈ ℕ 𝑓 = ( 𝐽 ∖ (𝑔𝑥))}
3329, 32eqtr4di 2791 . . . . . 6 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → 𝑥 ∈ ℕ ( 𝐽 ∖ (𝑔𝑥)) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3423, 33eqtr3d 2775 . . . . 5 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → ( 𝐽 ran 𝑔) = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
35 rneq 5779 . . . . . . 7 (𝑓 = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3635unieqd 4810 . . . . . 6 (𝑓 = (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))) → ran 𝑓 = ran (𝑥 ∈ ℕ ↦ ( 𝐽 ∖ (𝑔𝑥))))
3736rspceeqv 3541 . . . . 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)
3817, 34, 37syl2anc 587 . . . 4 ((𝐽 ∈ PNrm ∧ 𝑔 ∈ (𝐽m ℕ)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)( 𝐽 ran 𝑔) = ran 𝑓)
3938ad2ant2r 747 . . 3 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽m ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)( 𝐽 ran 𝑔) = ran 𝑓)
40 difeq2 4007 . . . . . . . 8 (( 𝐽𝐴) = ran 𝑔 → ( 𝐽 ∖ ( 𝐽𝐴)) = ( 𝐽 ran 𝑔))
4140eqcomd 2744 . . . . . . 7 (( 𝐽𝐴) = ran 𝑔 → ( 𝐽 ran 𝑔) = ( 𝐽 ∖ ( 𝐽𝐴)))
42 elssuni 4828 . . . . . . . 8 (𝐴𝐽𝐴 𝐽)
43 dfss4 4149 . . . . . . . 8 (𝐴 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝐴)) = 𝐴)
4442, 43sylib 221 . . . . . . 7 (𝐴𝐽 → ( 𝐽 ∖ ( 𝐽𝐴)) = 𝐴)
4541, 44sylan9eqr 2795 . . . . . 6 ((𝐴𝐽 ∧ ( 𝐽𝐴) = ran 𝑔) → ( 𝐽 ran 𝑔) = 𝐴)
4645ad2ant2l 746 . . . . 5 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽m ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ( 𝐽 ran 𝑔) = 𝐴)
4746eqeq1d 2740 . . . 4 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽m ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → (( 𝐽 ran 𝑔) = ran 𝑓𝐴 = ran 𝑓))
4847rexbidv 3207 . . 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))
4939, 48mpbid 235 . 2 (((𝐽 ∈ PNrm ∧ 𝐴𝐽) ∧ (𝑔 ∈ (𝐽m ℕ) ∧ ( 𝐽𝐴) = ran 𝑔)) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ran 𝑓)
506, 49rexlimddv 3201 1 ((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ran 𝑓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  {cab 2716  wral 3053  wrex 3054  Vcvv 3398  cdif 3840  wss 3843   cuni 4796   cint 4836   ciun 4881   ciin 4882  cmpt 5110  ran crn 5526   Fn wfn 6334  wf 6335  cfv 6339  (class class class)co 7170  m cmap 8437  cn 11716  Topctop 21644  Clsdccld 21767  PNrmcpnrm 22063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-1cn 10673  ax-addcl 10675
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-map 8439  df-nn 11717  df-top 21645  df-cld 21770  df-nrm 22068  df-pnrm 22070
This theorem is referenced by: (None)
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