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Mirrors > Home > MPE Home > Th. List > 2ndcrest | Structured version Visualization version GIF version |
Description: A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
2ndcrest | ⊢ ((𝐽 ∈ 2ndω ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ 2ndω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | is2ndc 22056 | . . 3 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) | |
2 | simplr 767 | . . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → 𝑥 ∈ TopBases) | |
3 | simpll 765 | . . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → 𝐴 ∈ 𝑉) | |
4 | tgrest 21769 | . . . . . . . 8 ⊢ ((𝑥 ∈ TopBases ∧ 𝐴 ∈ 𝑉) → (topGen‘(𝑥 ↾t 𝐴)) = ((topGen‘𝑥) ↾t 𝐴)) | |
5 | 2, 3, 4 | syl2anc 586 | . . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (topGen‘(𝑥 ↾t 𝐴)) = ((topGen‘𝑥) ↾t 𝐴)) |
6 | restbas 21768 | . . . . . . . . 9 ⊢ (𝑥 ∈ TopBases → (𝑥 ↾t 𝐴) ∈ TopBases) | |
7 | 6 | ad2antlr 725 | . . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥 ↾t 𝐴) ∈ TopBases) |
8 | restval 16702 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ TopBases ∧ 𝐴 ∈ 𝑉) → (𝑥 ↾t 𝐴) = ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ 𝐴))) | |
9 | 2, 3, 8 | syl2anc 586 | . . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥 ↾t 𝐴) = ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ 𝐴))) |
10 | 1stcrestlem 22062 | . . . . . . . . . 10 ⊢ (𝑥 ≼ ω → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ 𝐴)) ≼ ω) | |
11 | 10 | adantl 484 | . . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ran (𝑦 ∈ 𝑥 ↦ (𝑦 ∩ 𝐴)) ≼ ω) |
12 | 9, 11 | eqbrtrd 5090 | . . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥 ↾t 𝐴) ≼ ω) |
13 | 2ndci 22058 | . . . . . . . 8 ⊢ (((𝑥 ↾t 𝐴) ∈ TopBases ∧ (𝑥 ↾t 𝐴) ≼ ω) → (topGen‘(𝑥 ↾t 𝐴)) ∈ 2ndω) | |
14 | 7, 12, 13 | syl2anc 586 | . . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (topGen‘(𝑥 ↾t 𝐴)) ∈ 2ndω) |
15 | 5, 14 | eqeltrrd 2916 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ((topGen‘𝑥) ↾t 𝐴) ∈ 2ndω) |
16 | oveq1 7165 | . . . . . . 7 ⊢ ((topGen‘𝑥) = 𝐽 → ((topGen‘𝑥) ↾t 𝐴) = (𝐽 ↾t 𝐴)) | |
17 | 16 | eleq1d 2899 | . . . . . 6 ⊢ ((topGen‘𝑥) = 𝐽 → (((topGen‘𝑥) ↾t 𝐴) ∈ 2ndω ↔ (𝐽 ↾t 𝐴) ∈ 2ndω)) |
18 | 15, 17 | syl5ibcom 247 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ((topGen‘𝑥) = 𝐽 → (𝐽 ↾t 𝐴) ∈ 2ndω)) |
19 | 18 | expimpd 456 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases) → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (𝐽 ↾t 𝐴) ∈ 2ndω)) |
20 | 19 | rexlimdva 3286 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (𝐽 ↾t 𝐴) ∈ 2ndω)) |
21 | 1, 20 | syl5bi 244 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐽 ∈ 2ndω → (𝐽 ↾t 𝐴) ∈ 2ndω)) |
22 | 21 | impcom 410 | 1 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ 2ndω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 ∩ cin 3937 class class class wbr 5068 ↦ cmpt 5148 ran crn 5558 ‘cfv 6357 (class class class)co 7158 ωcom 7582 ≼ cdom 8509 ↾t crest 16696 topGenctg 16713 TopBasesctb 21555 2ndωc2ndc 22048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-fin 8515 df-fi 8877 df-card 9370 df-acn 9373 df-rest 16698 df-topgen 16719 df-bases 21556 df-2ndc 22050 |
This theorem is referenced by: (None) |
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