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Mirrors > Home > MPE Home > Th. List > fibas | Structured version Visualization version GIF version |
Description: A collection of finite intersections is a basis. The initial set is a subbasis for the topology. (Contributed by Jeff Hankins, 25-Aug-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
Ref | Expression |
---|---|
fibas | ⊢ (fi‘𝐴) ∈ TopBases |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6459 | . 2 ⊢ (fi‘𝐴) ∈ V | |
2 | fiin 8616 | . . 3 ⊢ ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥 ∩ 𝑦) ∈ (fi‘𝐴)) | |
3 | 2 | rgen2a 3158 | . 2 ⊢ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥 ∩ 𝑦) ∈ (fi‘𝐴) |
4 | fiinbas 21164 | . 2 ⊢ (((fi‘𝐴) ∈ V ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥 ∩ 𝑦) ∈ (fi‘𝐴)) → (fi‘𝐴) ∈ TopBases) | |
5 | 1, 3, 4 | mp2an 682 | 1 ⊢ (fi‘𝐴) ∈ TopBases |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ∀wral 3089 Vcvv 3397 ∩ cin 3790 ‘cfv 6135 ficfi 8604 TopBasesctb 21157 |
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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 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-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-oadd 7847 df-er 8026 df-en 8242 df-fin 8245 df-fi 8605 df-bases 21158 |
This theorem is referenced by: restbas 21370 ordttopon 21405 ordtopn1 21406 ordtopn2 21407 ordtrest2 21416 leordtval2 21424 2ndcsb 21661 ptbas 21791 xkotop 21800 alexsublem 22256 alexsub 22257 alexsubb 22258 alexsubALTlem3 22261 alexsubALTlem4 22262 alexsubALT 22263 ptcmplem1 22264 ordtrest2NEW 30567 topjoin 32948 |
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