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Mirrors > Home > MPE Home > Th. List > iscnrm2 | Structured version Visualization version GIF version |
Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
iscnrm2 | ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 21125 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
2 | eqid 2778 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | iscnrm 21535 | . . . 4 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm)) |
4 | 3 | baib 531 | . . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm)) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm)) |
6 | toponuni 21126 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
7 | 6 | pweqd 4384 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝒫 𝑋 = 𝒫 ∪ 𝐽) |
8 | 7 | raleqdv 3340 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm)) |
9 | 5, 8 | bitr4d 274 | 1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2107 ∀wral 3090 𝒫 cpw 4379 ∪ cuni 4671 ‘cfv 6135 (class class class)co 6922 ↾t crest 16467 Topctop 21105 TopOnctopon 21122 Nrmcnrm 21522 CNrmccnrm 21523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-topon 21123 df-cnrm 21530 |
This theorem is referenced by: restcnrm 21574 |
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