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Theorem iscnrm2 23241
Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
iscnrm2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ CNrm ↔ βˆ€π‘₯ ∈ 𝒫 𝑋(𝐽 β†Ύt π‘₯) ∈ Nrm))
Distinct variable groups:   π‘₯,𝐽   π‘₯,𝑋

Proof of Theorem iscnrm2
StepHypRef Expression
1 topontop 22814 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 eqid 2728 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
32iscnrm 23226 . . . 4 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝒫 βˆͺ 𝐽(𝐽 β†Ύt π‘₯) ∈ Nrm))
43baib 535 . . 3 (𝐽 ∈ Top β†’ (𝐽 ∈ CNrm ↔ βˆ€π‘₯ ∈ 𝒫 βˆͺ 𝐽(𝐽 β†Ύt π‘₯) ∈ Nrm))
51, 4syl 17 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ CNrm ↔ βˆ€π‘₯ ∈ 𝒫 βˆͺ 𝐽(𝐽 β†Ύt π‘₯) ∈ Nrm))
6 toponuni 22815 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
76pweqd 4620 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
87raleqdv 3322 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘₯ ∈ 𝒫 𝑋(𝐽 β†Ύt π‘₯) ∈ Nrm ↔ βˆ€π‘₯ ∈ 𝒫 βˆͺ 𝐽(𝐽 β†Ύt π‘₯) ∈ Nrm))
95, 8bitr4d 282 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ CNrm ↔ βˆ€π‘₯ ∈ 𝒫 𝑋(𝐽 β†Ύt π‘₯) ∈ Nrm))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∈ wcel 2099  βˆ€wral 3058  π’« cpw 4603  βˆͺ cuni 4908  β€˜cfv 6548  (class class class)co 7420   β†Ύt crest 17401  Topctop 22794  TopOnctopon 22811  Nrmcnrm 23213  CNrmccnrm 23214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-topon 22812  df-cnrm 23221
This theorem is referenced by:  restcnrm  23265
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