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Mirrors > Home > HSE Home > Th. List > chintcli | Structured version Visualization version GIF version |
Description: The intersection of a nonempty set of closed subspaces is a closed subspace. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chintcl.1 | ⊢ (𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅) |
Ref | Expression |
---|---|
chintcli | ⊢ ∩ 𝐴 ∈ Cℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chintcl.1 | . . . . . 6 ⊢ (𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅) | |
2 | 1 | simpli 483 | . . . . 5 ⊢ 𝐴 ⊆ Cℋ |
3 | chsssh 31248 | . . . . 5 ⊢ Cℋ ⊆ Sℋ | |
4 | 2, 3 | sstri 4012 | . . . 4 ⊢ 𝐴 ⊆ Sℋ |
5 | 1 | simpri 485 | . . . 4 ⊢ 𝐴 ≠ ∅ |
6 | 4, 5 | pm3.2i 470 | . . 3 ⊢ (𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅) |
7 | 6 | shintcli 31352 | . 2 ⊢ ∩ 𝐴 ∈ Sℋ |
8 | 2 | sseli 3998 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ Cℋ ) |
9 | vex 3486 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
10 | 9 | chlimi 31257 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ Cℋ ∧ 𝑓:ℕ⟶𝑦 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝑦) |
11 | 10 | 3exp 1119 | . . . . . . . . 9 ⊢ (𝑦 ∈ Cℋ → (𝑓:ℕ⟶𝑦 → (𝑓 ⇝𝑣 𝑥 → 𝑥 ∈ 𝑦))) |
12 | 11 | com3r 87 | . . . . . . . 8 ⊢ (𝑓 ⇝𝑣 𝑥 → (𝑦 ∈ Cℋ → (𝑓:ℕ⟶𝑦 → 𝑥 ∈ 𝑦))) |
13 | 8, 12 | syl5 34 | . . . . . . 7 ⊢ (𝑓 ⇝𝑣 𝑥 → (𝑦 ∈ 𝐴 → (𝑓:ℕ⟶𝑦 → 𝑥 ∈ 𝑦))) |
14 | 13 | imp 406 | . . . . . 6 ⊢ ((𝑓 ⇝𝑣 𝑥 ∧ 𝑦 ∈ 𝐴) → (𝑓:ℕ⟶𝑦 → 𝑥 ∈ 𝑦)) |
15 | 14 | ralimdva 3169 | . . . . 5 ⊢ (𝑓 ⇝𝑣 𝑥 → (∀𝑦 ∈ 𝐴 𝑓:ℕ⟶𝑦 → ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
16 | 5 | fint 6799 | . . . . 5 ⊢ (𝑓:ℕ⟶∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑓:ℕ⟶𝑦) |
17 | 9 | elint2 4979 | . . . . 5 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
18 | 15, 16, 17 | 3imtr4g 296 | . . . 4 ⊢ (𝑓 ⇝𝑣 𝑥 → (𝑓:ℕ⟶∩ 𝐴 → 𝑥 ∈ ∩ 𝐴)) |
19 | 18 | impcom 407 | . . 3 ⊢ ((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴) |
20 | 19 | gen2 1794 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴) |
21 | isch2 31246 | . 2 ⊢ (∩ 𝐴 ∈ Cℋ ↔ (∩ 𝐴 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴))) | |
22 | 7, 20, 21 | mpbir2an 710 | 1 ⊢ ∩ 𝐴 ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 ∈ wcel 2103 ≠ wne 2942 ∀wral 3063 ⊆ wss 3970 ∅c0 4347 ∩ cint 4972 class class class wbr 5169 ⟶wf 6568 ℕcn 12289 ⇝𝑣 chli 30950 Sℋ csh 30951 Cℋ cch 30952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-1cn 11238 ax-addcl 11240 ax-hilex 31022 ax-hfvadd 31023 ax-hv0cl 31026 ax-hfvmul 31028 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-map 8882 df-nn 12290 df-sh 31230 df-ch 31244 |
This theorem is referenced by: chintcl 31355 chincli 31483 |
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