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Theorem cnrmi 21158
 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 2621 . . 3 𝐽 = 𝐽
21restin 20964 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) = (𝐽t (𝐴 𝐽)))
3 inss2 3832 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
4 inex1g 4799 . . . . . 6 (𝐴𝑉 → (𝐴 𝐽) ∈ V)
5 elpwg 4164 . . . . . 6 ((𝐴 𝐽) ∈ V → ((𝐴 𝐽) ∈ 𝒫 𝐽 ↔ (𝐴 𝐽) ⊆ 𝐽))
64, 5syl 17 . . . . 5 (𝐴𝑉 → ((𝐴 𝐽) ∈ 𝒫 𝐽 ↔ (𝐴 𝐽) ⊆ 𝐽))
73, 6mpbiri 248 . . . 4 (𝐴𝑉 → (𝐴 𝐽) ∈ 𝒫 𝐽)
87adantl 482 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐴 𝐽) ∈ 𝒫 𝐽)
91iscnrm 21121 . . . . 5 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
109simprbi 480 . . . 4 (𝐽 ∈ CNrm → ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm)
1110adantr 481 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm)
12 oveq2 6655 . . . . 5 (𝑥 = (𝐴 𝐽) → (𝐽t 𝑥) = (𝐽t (𝐴 𝐽)))
1312eleq1d 2685 . . . 4 (𝑥 = (𝐴 𝐽) → ((𝐽t 𝑥) ∈ Nrm ↔ (𝐽t (𝐴 𝐽)) ∈ Nrm))
1413rspcv 3303 . . 3 ((𝐴 𝐽) ∈ 𝒫 𝐽 → (∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm → (𝐽t (𝐴 𝐽)) ∈ Nrm))
158, 11, 14sylc 65 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t (𝐴 𝐽)) ∈ Nrm)
162, 15eqeltrd 2700 1 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Nrm)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1482   ∈ wcel 1989  ∀wral 2911  Vcvv 3198   ∩ cin 3571   ⊆ wss 3572  𝒫 cpw 4156  ∪ cuni 4434  (class class class)co 6647   ↾t crest 16075  Topctop 20692  Nrmcnrm 21108  CNrmccnrm 21109 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-rest 16077  df-cnrm 21116 This theorem is referenced by:  cnrmnrm  21159  restcnrm  21160
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