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Theorem cnrmi 21954
Description: A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
cnrmi ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Nrm)

Proof of Theorem cnrmi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . 3 𝐽 = 𝐽
21restin 21760 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) = (𝐽t (𝐴 𝐽)))
3 oveq2 7146 . . . 4 (𝑥 = (𝐴 𝐽) → (𝐽t 𝑥) = (𝐽t (𝐴 𝐽)))
43eleq1d 2900 . . 3 (𝑥 = (𝐴 𝐽) → ((𝐽t 𝑥) ∈ Nrm ↔ (𝐽t (𝐴 𝐽)) ∈ Nrm))
51iscnrm 21917 . . . . 5 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
65simprbi 500 . . . 4 (𝐽 ∈ CNrm → ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm)
76adantr 484 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm)
8 inss2 4189 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
9 inex1g 5204 . . . . . 6 (𝐴𝑉 → (𝐴 𝐽) ∈ V)
10 elpwg 4523 . . . . . 6 ((𝐴 𝐽) ∈ V → ((𝐴 𝐽) ∈ 𝒫 𝐽 ↔ (𝐴 𝐽) ⊆ 𝐽))
119, 10syl 17 . . . . 5 (𝐴𝑉 → ((𝐴 𝐽) ∈ 𝒫 𝐽 ↔ (𝐴 𝐽) ⊆ 𝐽))
128, 11mpbiri 261 . . . 4 (𝐴𝑉 → (𝐴 𝐽) ∈ 𝒫 𝐽)
1312adantl 485 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐴 𝐽) ∈ 𝒫 𝐽)
144, 7, 13rspcdva 3610 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t (𝐴 𝐽)) ∈ Nrm)
152, 14eqeltrd 2916 1 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3132  Vcvv 3479  cin 3917  wss 3918  𝒫 cpw 4520   cuni 4819  (class class class)co 7138  t crest 16683  Topctop 21487  Nrmcnrm 21904  CNrmccnrm 21905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-rest 16685  df-cnrm 21912
This theorem is referenced by:  cnrmnrm  21955  restcnrm  21956
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