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Theorem iscnrm 23347
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 4923 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ist0.1 . . . . 5 𝑋 = 𝐽
31, 2eqtr4di 2793 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
43pweqd 4622 . . 3 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
5 oveq1 7438 . . . 4 (𝑗 = 𝐽 → (𝑗t 𝑥) = (𝐽t 𝑥))
65eleq1d 2824 . . 3 (𝑗 = 𝐽 → ((𝑗t 𝑥) ∈ Nrm ↔ (𝐽t 𝑥) ∈ Nrm))
74, 6raleqbidv 3344 . 2 (𝑗 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
8 df-cnrm 23342 . 2 CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm}
97, 8elrab2 3698 1 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  𝒫 cpw 4605   cuni 4912  (class class class)co 7431  t crest 17467  Topctop 22915  Nrmcnrm 23334  CNrmccnrm 23335
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-cnrm 23342
This theorem is referenced by:  cnrmtop  23361  iscnrm2  23362  cnrmi  23384  iscnrm3  48749
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