MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscnrm2 Structured version   Visualization version   GIF version

Theorem iscnrm2 23346
Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
iscnrm2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem iscnrm2
StepHypRef Expression
1 topontop 22919 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 eqid 2737 . . . . 5 𝐽 = 𝐽
32iscnrm 23331 . . . 4 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
43baib 535 . . 3 (𝐽 ∈ Top → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
51, 4syl 17 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
6 toponuni 22920 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76pweqd 4617 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝒫 𝑋 = 𝒫 𝐽)
87raleqdv 3326 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
95, 8bitr4d 282 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  wral 3061  𝒫 cpw 4600   cuni 4907  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  TopOnctopon 22916  Nrmcnrm 23318  CNrmccnrm 23319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-topon 22917  df-cnrm 23326
This theorem is referenced by:  restcnrm  23370
  Copyright terms: Public domain W3C validator