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Theorem cnrmi 22734
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 2733 . . 3 𝐽 = 𝐽
21restin 22540 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) = (𝐽t (𝐴 𝐽)))
3 oveq2 7369 . . . 4 (𝑥 = (𝐴 𝐽) → (𝐽t 𝑥) = (𝐽t (𝐴 𝐽)))
43eleq1d 2819 . . 3 (𝑥 = (𝐴 𝐽) → ((𝐽t 𝑥) ∈ Nrm ↔ (𝐽t (𝐴 𝐽)) ∈ Nrm))
51iscnrm 22697 . . . . 5 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
65simprbi 498 . . . 4 (𝐽 ∈ CNrm → ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm)
76adantr 482 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm)
8 inss2 4193 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
9 inex1g 5280 . . . . . 6 (𝐴𝑉 → (𝐴 𝐽) ∈ V)
10 elpwg 4567 . . . . . 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 258 . . . 4 (𝐴𝑉 → (𝐴 𝐽) ∈ 𝒫 𝐽)
1312adantl 483 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐴 𝐽) ∈ 𝒫 𝐽)
144, 7, 13rspcdva 3584 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t (𝐴 𝐽)) ∈ Nrm)
152, 14eqeltrd 2834 1 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  Vcvv 3447  cin 3913  wss 3914  𝒫 cpw 4564   cuni 4869  (class class class)co 7361  t crest 17310  Topctop 22265  Nrmcnrm 22684  CNrmccnrm 22685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-rest 17312  df-cnrm 22692
This theorem is referenced by:  cnrmnrm  22735  restcnrm  22736
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