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Theorem iscnrm2 23186
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 22759 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 eqid 2724 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
32iscnrm 23171 . . . 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 22760 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
76pweqd 4612 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
87raleqdv 3317 . 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 2098  βˆ€wral 3053  π’« cpw 4595  βˆͺ cuni 4900  β€˜cfv 6534  (class class class)co 7402   β†Ύt crest 17371  Topctop 22739  TopOnctopon 22756  Nrmcnrm 23158  CNrmccnrm 23159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-topon 22757  df-cnrm 23166
This theorem is referenced by:  restcnrm  23210
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