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Theorem iscnrm 22076
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 4807 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ist0.1 . . . . 5 𝑋 = 𝐽
31, 2eqtr4di 2791 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
43pweqd 4507 . . 3 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
5 oveq1 7179 . . . 4 (𝑗 = 𝐽 → (𝑗t 𝑥) = (𝐽t 𝑥))
65eleq1d 2817 . . 3 (𝑗 = 𝐽 → ((𝑗t 𝑥) ∈ Nrm ↔ (𝐽t 𝑥) ∈ Nrm))
74, 6raleqbidv 3304 . 2 (𝑗 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
8 df-cnrm 22071 . 2 CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm}
97, 8elrab2 3591 1 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1542  wcel 2114  wral 3053  𝒫 cpw 4488   cuni 4796  (class class class)co 7172  t crest 16799  Topctop 21646  Nrmcnrm 22063  CNrmccnrm 22064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rab 3062  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-iota 6297  df-fv 6347  df-ov 7175  df-cnrm 22071
This theorem is referenced by:  cnrmtop  22090  iscnrm2  22091  cnrmi  22113  iscnrm3  45797
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