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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 478 | . . . . 5 ⊢ 𝐴 ⊆ Cℋ |
3 | chsssh 28637 | . . . . 5 ⊢ Cℋ ⊆ Sℋ | |
4 | 2, 3 | sstri 3836 | . . . 4 ⊢ 𝐴 ⊆ Sℋ |
5 | 1 | simpri 481 | . . . 4 ⊢ 𝐴 ≠ ∅ |
6 | 4, 5 | pm3.2i 464 | . . 3 ⊢ (𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅) |
7 | 6 | shintcli 28743 | . 2 ⊢ ∩ 𝐴 ∈ Sℋ |
8 | 2 | sseli 3823 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ Cℋ ) |
9 | vex 3417 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
10 | 9 | chlimi 28646 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ Cℋ ∧ 𝑓:ℕ⟶𝑦 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝑦) |
11 | 10 | 3exp 1154 | . . . . . . . . 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 397 | . . . . . 6 ⊢ ((𝑓 ⇝𝑣 𝑥 ∧ 𝑦 ∈ 𝐴) → (𝑓:ℕ⟶𝑦 → 𝑥 ∈ 𝑦)) |
15 | 14 | ralimdva 3171 | . . . . 5 ⊢ (𝑓 ⇝𝑣 𝑥 → (∀𝑦 ∈ 𝐴 𝑓:ℕ⟶𝑦 → ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
16 | 5 | fint 6321 | . . . . 5 ⊢ (𝑓:ℕ⟶∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑓:ℕ⟶𝑦) |
17 | 9 | elint2 4704 | . . . . 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 398 | . . 3 ⊢ ((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴) |
20 | 19 | gen2 1897 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴) |
21 | isch2 28635 | . 2 ⊢ (∩ 𝐴 ∈ Cℋ ↔ (∩ 𝐴 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴))) | |
22 | 7, 20, 21 | mpbir2an 704 | 1 ⊢ ∩ 𝐴 ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∀wal 1656 ∈ wcel 2166 ≠ wne 2999 ∀wral 3117 ⊆ wss 3798 ∅c0 4144 ∩ cint 4697 class class class wbr 4873 ⟶wf 6119 ℕcn 11350 ⇝𝑣 chli 28339 Sℋ csh 28340 Cℋ cch 28341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-1cn 10310 ax-addcl 10312 ax-hilex 28411 ax-hfvadd 28412 ax-hv0cl 28415 ax-hfvmul 28417 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-map 8124 df-nn 11351 df-sh 28619 df-ch 28633 |
This theorem is referenced by: chintcl 28746 chincli 28874 |
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