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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 6895 | . 2 ⊢ (fi‘𝐴) ∈ V | |
2 | fiin 9414 | . . 3 ⊢ ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥 ∩ 𝑦) ∈ (fi‘𝐴)) | |
3 | 2 | rgen2 3189 | . 2 ⊢ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥 ∩ 𝑦) ∈ (fi‘𝐴) |
4 | fiinbas 22799 | . 2 ⊢ (((fi‘𝐴) ∈ V ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥 ∩ 𝑦) ∈ (fi‘𝐴)) → (fi‘𝐴) ∈ TopBases) | |
5 | 1, 3, 4 | mp2an 689 | 1 ⊢ (fi‘𝐴) ∈ TopBases |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∀wral 3053 Vcvv 3466 ∩ cin 3940 ‘cfv 6534 ficfi 9402 TopBasesctb 22792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-om 7850 df-en 8937 df-fin 8940 df-fi 9403 df-bases 22793 |
This theorem is referenced by: restbas 23006 ordttopon 23041 ordtopn1 23042 ordtopn2 23043 ordtrest2 23052 leordtval2 23060 2ndcsb 23297 ptbas 23427 xkotop 23436 alexsublem 23892 alexsub 23893 alexsubb 23894 alexsubALTlem3 23897 alexsubALTlem4 23898 alexsubALT 23899 ptcmplem1 23900 ordtrest2NEW 33422 topjoin 35750 |
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