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Theorem iscnrm2 21941
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 21516 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 eqid 2822 . . . . 5 𝐽 = 𝐽
32iscnrm 21926 . . . 4 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
43baib 539 . . 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 21517 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76pweqd 4530 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝒫 𝑋 = 𝒫 𝐽)
87raleqdv 3392 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
95, 8bitr4d 285 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2114  wral 3130  𝒫 cpw 4511   cuni 4813  cfv 6334  (class class class)co 7140  t crest 16685  Topctop 21496  TopOnctopon 21513  Nrmcnrm 21913  CNrmccnrm 21914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-topon 21514  df-cnrm 21921
This theorem is referenced by:  restcnrm  21965
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