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Theorem iscnrm 22502
Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypothesis
Ref Expression
ist0.1 𝑋 = 𝐽
Assertion
Ref Expression
iscnrm (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem iscnrm
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 unieq 4852 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ist0.1 . . . . 5 𝑋 = 𝐽
31, 2eqtr4di 2791 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
43pweqd 4555 . . 3 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
5 oveq1 7302 . . . 4 (𝑗 = 𝐽 → (𝑗t 𝑥) = (𝐽t 𝑥))
65eleq1d 2818 . . 3 (𝑗 = 𝐽 → ((𝑗t 𝑥) ∈ Nrm ↔ (𝐽t 𝑥) ∈ Nrm))
74, 6raleqbidv 3338 . 2 (𝑗 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
8 df-cnrm 22497 . 2 CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm}
97, 8elrab2 3629 1 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1537  wcel 2101  wral 3059  𝒫 cpw 4536   cuni 4841  (class class class)co 7295  t crest 17159  Topctop 22070  Nrmcnrm 22489  CNrmccnrm 22490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3060  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-iota 6399  df-fv 6455  df-ov 7298  df-cnrm 22497
This theorem is referenced by:  cnrmtop  22516  iscnrm2  22517  cnrmi  22539  iscnrm3  46286
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