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Mirrors > Home > MPE Home > Th. List > cnrmnrm | Structured version Visualization version GIF version |
Description: A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
cnrmnrm | ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Nrm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | restid 16701 | . 2 ⊢ (𝐽 ∈ CNrm → (𝐽 ↾t ∪ 𝐽) = 𝐽) |
3 | uniexg 7460 | . . 3 ⊢ (𝐽 ∈ CNrm → ∪ 𝐽 ∈ V) | |
4 | cnrmi 21962 | . . 3 ⊢ ((𝐽 ∈ CNrm ∧ ∪ 𝐽 ∈ V) → (𝐽 ↾t ∪ 𝐽) ∈ Nrm) | |
5 | 3, 4 | mpdan 685 | . 2 ⊢ (𝐽 ∈ CNrm → (𝐽 ↾t ∪ 𝐽) ∈ Nrm) |
6 | 2, 5 | eqeltrrd 2914 | 1 ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Nrm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3494 ∪ cuni 4831 (class class class)co 7150 ↾t crest 16688 Nrmcnrm 21912 CNrmccnrm 21913 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-rest 16690 df-cnrm 21920 |
This theorem is referenced by: (None) |
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