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Theorem iscnrm 23214
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 4914 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ist0.1 . . . . 5 𝑋 = 𝐽
31, 2eqtr4di 2785 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
43pweqd 4615 . . 3 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
5 oveq1 7421 . . . 4 (𝑗 = 𝐽 → (𝑗t 𝑥) = (𝐽t 𝑥))
65eleq1d 2813 . . 3 (𝑗 = 𝐽 → ((𝑗t 𝑥) ∈ Nrm ↔ (𝐽t 𝑥) ∈ Nrm))
74, 6raleqbidv 3337 . 2 (𝑗 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
8 df-cnrm 23209 . 2 CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm}
97, 8elrab2 3683 1 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  wcel 2099  wral 3056  𝒫 cpw 4598   cuni 4903  (class class class)co 7414  t crest 17393  Topctop 22782  Nrmcnrm 23201  CNrmccnrm 23202
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-ext 2698
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-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-cnrm 23209
This theorem is referenced by:  cnrmtop  23228  iscnrm2  23229  cnrmi  23251  iscnrm3  47894
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