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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 482 | . . . . 5 ⊢ 𝐴 ⊆ Cℋ |
3 | chsssh 31063 | . . . . 5 ⊢ Cℋ ⊆ Sℋ | |
4 | 2, 3 | sstri 3991 | . . . 4 ⊢ 𝐴 ⊆ Sℋ |
5 | 1 | simpri 484 | . . . 4 ⊢ 𝐴 ≠ ∅ |
6 | 4, 5 | pm3.2i 469 | . . 3 ⊢ (𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅) |
7 | 6 | shintcli 31167 | . 2 ⊢ ∩ 𝐴 ∈ Sℋ |
8 | 2 | sseli 3978 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ Cℋ ) |
9 | vex 3477 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
10 | 9 | chlimi 31072 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ Cℋ ∧ 𝑓:ℕ⟶𝑦 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝑦) |
11 | 10 | 3exp 1116 | . . . . . . . . 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 405 | . . . . . 6 ⊢ ((𝑓 ⇝𝑣 𝑥 ∧ 𝑦 ∈ 𝐴) → (𝑓:ℕ⟶𝑦 → 𝑥 ∈ 𝑦)) |
15 | 14 | ralimdva 3164 | . . . . 5 ⊢ (𝑓 ⇝𝑣 𝑥 → (∀𝑦 ∈ 𝐴 𝑓:ℕ⟶𝑦 → ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
16 | 5 | fint 6781 | . . . . 5 ⊢ (𝑓:ℕ⟶∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑓:ℕ⟶𝑦) |
17 | 9 | elint2 4960 | . . . . 5 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
18 | 15, 16, 17 | 3imtr4g 295 | . . . 4 ⊢ (𝑓 ⇝𝑣 𝑥 → (𝑓:ℕ⟶∩ 𝐴 → 𝑥 ∈ ∩ 𝐴)) |
19 | 18 | impcom 406 | . . 3 ⊢ ((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴) |
20 | 19 | gen2 1790 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴) |
21 | isch2 31061 | . 2 ⊢ (∩ 𝐴 ∈ Cℋ ↔ (∩ 𝐴 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴))) | |
22 | 7, 20, 21 | mpbir2an 709 | 1 ⊢ ∩ 𝐴 ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1531 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ⊆ wss 3949 ∅c0 4326 ∩ cint 4953 class class class wbr 5152 ⟶wf 6549 ℕcn 12252 ⇝𝑣 chli 30765 Sℋ csh 30766 Cℋ cch 30767 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-1cn 11206 ax-addcl 11208 ax-hilex 30837 ax-hfvadd 30838 ax-hv0cl 30841 ax-hfvmul 30843 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-map 8855 df-nn 12253 df-sh 31045 df-ch 31059 |
This theorem is referenced by: chintcl 31170 chincli 31298 |
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