Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > riinopn | Structured version Visualization version GIF version |
Description: A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
1open.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
riinopn | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riin0 5006 | . . . 4 ⊢ (𝐴 = ∅ → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = 𝑋) | |
2 | 1 | adantl 484 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) ∧ 𝐴 = ∅) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = 𝑋) |
3 | simpl1 1187 | . . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) ∧ 𝐴 = ∅) → 𝐽 ∈ Top) | |
4 | 1open.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
5 | 4 | topopn 21516 | . . . 4 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
6 | 3, 5 | syl 17 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) ∧ 𝐴 = ∅) → 𝑋 ∈ 𝐽) |
7 | 2, 6 | eqeltrd 2915 | . 2 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) ∧ 𝐴 = ∅) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ 𝐽) |
8 | 4 | eltopss 21517 | . . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → 𝐵 ⊆ 𝑋) |
9 | 8 | ex 415 | . . . . . . 7 ⊢ (𝐽 ∈ Top → (𝐵 ∈ 𝐽 → 𝐵 ⊆ 𝑋)) |
10 | 9 | adantr 483 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐵 ∈ 𝐽 → 𝐵 ⊆ 𝑋)) |
11 | 10 | ralimdv 3180 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝑋)) |
12 | 11 | 3impia 1113 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝑋) |
13 | riinn0 5007 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝑋 ∧ 𝐴 ≠ ∅) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 𝐵) | |
14 | 12, 13 | sylan 582 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) ∧ 𝐴 ≠ ∅) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 𝐵) |
15 | iinopn 21512 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) | |
16 | 15 | 3exp2 1350 | . . . . 5 ⊢ (𝐽 ∈ Top → (𝐴 ∈ Fin → (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)))) |
17 | 16 | com34 91 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 → (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)))) |
18 | 17 | 3imp1 1343 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) |
19 | 14, 18 | eqeltrd 2915 | . 2 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) ∧ 𝐴 ≠ ∅) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ 𝐽) |
20 | 7, 19 | pm2.61dane 3106 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 ∪ cuni 4840 ∩ ciin 4922 Fincfn 8511 Topctop 21503 |
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-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-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-iin 4924 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-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-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-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-fin 8515 df-top 21504 |
This theorem is referenced by: rintopn 21519 iuncld 21655 |
Copyright terms: Public domain | W3C validator |