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| 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 6918 | . 2 ⊢ (fi‘𝐴) ∈ V | |
| 2 | fiin 9463 | . . 3 ⊢ ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥 ∩ 𝑦) ∈ (fi‘𝐴)) | |
| 3 | 2 | rgen2 3198 | . 2 ⊢ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥 ∩ 𝑦) ∈ (fi‘𝐴) | 
| 4 | fiinbas 22960 | . 2 ⊢ (((fi‘𝐴) ∈ V ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥 ∩ 𝑦) ∈ (fi‘𝐴)) → (fi‘𝐴) ∈ TopBases) | |
| 5 | 1, 3, 4 | mp2an 692 | 1 ⊢ (fi‘𝐴) ∈ TopBases | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ∩ cin 3949 ‘cfv 6560 ficfi 9451 TopBasesctb 22953 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-om 7889 df-en 8987 df-fin 8990 df-fi 9452 df-bases 22954 | 
| This theorem is referenced by: restbas 23167 ordttopon 23202 ordtopn1 23203 ordtopn2 23204 ordtrest2 23213 leordtval2 23221 2ndcsb 23458 ptbas 23588 xkotop 23597 alexsublem 24053 alexsub 24054 alexsubb 24055 alexsubALTlem3 24058 alexsubALTlem4 24059 alexsubALT 24060 ptcmplem1 24061 ordtrest2NEW 33923 topjoin 36367 | 
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