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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 477 | . . . . 5 ⊢ 𝐴 ⊆ Cℋ |
3 | chsssh 28607 | . . . . 5 ⊢ Cℋ ⊆ Sℋ | |
4 | 2, 3 | sstri 3807 | . . . 4 ⊢ 𝐴 ⊆ Sℋ |
5 | 1 | simpri 480 | . . . 4 ⊢ 𝐴 ≠ ∅ |
6 | 4, 5 | pm3.2i 463 | . . 3 ⊢ (𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅) |
7 | 6 | shintcli 28713 | . 2 ⊢ ∩ 𝐴 ∈ Sℋ |
8 | 2 | sseli 3794 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ Cℋ ) |
9 | vex 3388 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
10 | 9 | chlimi 28616 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ Cℋ ∧ 𝑓:ℕ⟶𝑦 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝑦) |
11 | 10 | 3exp 1149 | . . . . . . . . 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 396 | . . . . . 6 ⊢ ((𝑓 ⇝𝑣 𝑥 ∧ 𝑦 ∈ 𝐴) → (𝑓:ℕ⟶𝑦 → 𝑥 ∈ 𝑦)) |
15 | 14 | ralimdva 3143 | . . . . 5 ⊢ (𝑓 ⇝𝑣 𝑥 → (∀𝑦 ∈ 𝐴 𝑓:ℕ⟶𝑦 → ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
16 | 5 | fint 6299 | . . . . 5 ⊢ (𝑓:ℕ⟶∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑓:ℕ⟶𝑦) |
17 | 9 | elint2 4674 | . . . . 5 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
18 | 15, 16, 17 | 3imtr4g 288 | . . . 4 ⊢ (𝑓 ⇝𝑣 𝑥 → (𝑓:ℕ⟶∩ 𝐴 → 𝑥 ∈ ∩ 𝐴)) |
19 | 18 | impcom 397 | . . 3 ⊢ ((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴) |
20 | 19 | gen2 1892 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴) |
21 | isch2 28605 | . 2 ⊢ (∩ 𝐴 ∈ Cℋ ↔ (∩ 𝐴 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴))) | |
22 | 7, 20, 21 | mpbir2an 703 | 1 ⊢ ∩ 𝐴 ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∀wal 1651 ∈ wcel 2157 ≠ wne 2971 ∀wral 3089 ⊆ wss 3769 ∅c0 4115 ∩ cint 4667 class class class wbr 4843 ⟶wf 6097 ℕcn 11312 ⇝𝑣 chli 28309 Sℋ csh 28310 Cℋ cch 28311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-1cn 10282 ax-addcl 10284 ax-hilex 28381 ax-hfvadd 28382 ax-hv0cl 28385 ax-hfvmul 28387 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-map 8097 df-nn 11313 df-sh 28589 df-ch 28603 |
This theorem is referenced by: chintcl 28716 chincli 28844 |
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