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Mirrors > Home > MPE Home > Th. List > rintopn | Structured version Visualization version GIF version |
Description: A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
1open.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
rintopn | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽 ∧ 𝐴 ∈ Fin) → (𝑋 ∩ ∩ 𝐴) ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intiin 4982 | . . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
2 | 1 | ineq2i 4185 | . 2 ⊢ (𝑋 ∩ ∩ 𝐴) = (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝑥) |
3 | dfss3 3955 | . . 3 ⊢ (𝐴 ⊆ 𝐽 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐽) | |
4 | 1open.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
5 | 4 | riinopn 21515 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐽) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝑥) ∈ 𝐽) |
6 | 5 | 3com23 1122 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐽 ∧ 𝐴 ∈ Fin) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝑥) ∈ 𝐽) |
7 | 3, 6 | syl3an2b 1400 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽 ∧ 𝐴 ∈ Fin) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝑥) ∈ 𝐽) |
8 | 2, 7 | eqeltrid 2917 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽 ∧ 𝐴 ∈ Fin) → (𝑋 ∩ ∩ 𝐴) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∩ cin 3934 ⊆ wss 3935 ∪ cuni 4837 ∩ cint 4875 ∩ ciin 4919 Fincfn 8508 Topctop 21500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-fin 8512 df-top 21501 |
This theorem is referenced by: ptcnplem 22228 tmdgsum2 22703 limciun 24491 |
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