MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscnrm Structured version   Visualization version   GIF version

Theorem iscnrm 21930
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 4848 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ist0.1 . . . . 5 𝑋 = 𝐽
31, 2syl6eqr 2874 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
43pweqd 4557 . . 3 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
5 oveq1 7162 . . . 4 (𝑗 = 𝐽 → (𝑗t 𝑥) = (𝐽t 𝑥))
65eleq1d 2897 . . 3 (𝑗 = 𝐽 → ((𝑗t 𝑥) ∈ Nrm ↔ (𝐽t 𝑥) ∈ Nrm))
74, 6raleqbidv 3401 . 2 (𝑗 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
8 df-cnrm 21925 . 2 CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm}
97, 8elrab2 3682 1 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1533  wcel 2110  wral 3138  𝒫 cpw 4538   cuni 4837  (class class class)co 7155  t crest 16693  Topctop 21500  Nrmcnrm 21917  CNrmccnrm 21918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-iota 6313  df-fv 6362  df-ov 7158  df-cnrm 21925
This theorem is referenced by:  cnrmtop  21944  iscnrm2  21945  cnrmi  21967
  Copyright terms: Public domain W3C validator