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Theorem iscnrm2 21550
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 21125 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 eqid 2778 . . . . 5 𝐽 = 𝐽
32iscnrm 21535 . . . 4 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
43baib 531 . . 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 21126 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76pweqd 4384 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝒫 𝑋 = 𝒫 𝐽)
87raleqdv 3340 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
95, 8bitr4d 274 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2107  wral 3090  𝒫 cpw 4379   cuni 4671  cfv 6135  (class class class)co 6922  t crest 16467  Topctop 21105  TopOnctopon 21122  Nrmcnrm 21522  CNrmccnrm 21523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-topon 21123  df-cnrm 21530
This theorem is referenced by:  restcnrm  21574
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