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Mirrors > Home > MPE Home > Th. List > hausmapdom | Structured version Visualization version GIF version |
Description: If 𝑋 is a first-countable Hausdorff space, then the cardinality of the closure of a set 𝐴 is bounded by ℕ to the power 𝐴. In particular, a first-countable Hausdorff space with a dense subset 𝐴 has cardinality at most 𝐴↑ℕ, and a separable first-countable Hausdorff space has cardinality at most 𝒫 ℕ. (Compare hauspwpwdom 23713 to see a weaker result if the assumption of first-countability is omitted.) (Contributed by Mario Carneiro, 9-Apr-2015.) |
Ref | Expression |
---|---|
hauspwdom.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
hausmapdom | ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ (𝐴 ↑m ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hauspwdom.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | 1stcelcls 23186 | . . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ ∃𝑓(𝑓:ℕ⟶𝐴 ∧ 𝑓(⇝𝑡‘𝐽)𝑥))) |
3 | 2 | 3adant1 1129 | . . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ ∃𝑓(𝑓:ℕ⟶𝐴 ∧ 𝑓(⇝𝑡‘𝐽)𝑥))) |
4 | uniexg 7733 | . . . . . . . . . . . 12 ⊢ (𝐽 ∈ Haus → ∪ 𝐽 ∈ V) | |
5 | 4 | 3ad2ant1 1132 | . . . . . . . . . . 11 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ∪ 𝐽 ∈ V) |
6 | 1, 5 | eqeltrid 2836 | . . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ V) |
7 | simp3 1137 | . . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) | |
8 | 6, 7 | ssexd 5325 | . . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
9 | nnex 12223 | . . . . . . . . 9 ⊢ ℕ ∈ V | |
10 | elmapg 8836 | . . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ℕ ∈ V) → (𝑓 ∈ (𝐴 ↑m ℕ) ↔ 𝑓:ℕ⟶𝐴)) | |
11 | 8, 9, 10 | sylancl 585 | . . . . . . . 8 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (𝑓 ∈ (𝐴 ↑m ℕ) ↔ 𝑓:ℕ⟶𝐴)) |
12 | 11 | anbi1d 629 | . . . . . . 7 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ((𝑓 ∈ (𝐴 ↑m ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥) ↔ (𝑓:ℕ⟶𝐴 ∧ 𝑓(⇝𝑡‘𝐽)𝑥))) |
13 | 12 | exbidv 1923 | . . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (∃𝑓(𝑓 ∈ (𝐴 ↑m ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥) ↔ ∃𝑓(𝑓:ℕ⟶𝐴 ∧ 𝑓(⇝𝑡‘𝐽)𝑥))) |
14 | 3, 13 | bitr4d 281 | . . . . 5 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ ∃𝑓(𝑓 ∈ (𝐴 ↑m ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥))) |
15 | df-rex 3070 | . . . . 5 ⊢ (∃𝑓 ∈ (𝐴 ↑m ℕ)𝑓(⇝𝑡‘𝐽)𝑥 ↔ ∃𝑓(𝑓 ∈ (𝐴 ↑m ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥)) | |
16 | 14, 15 | bitr4di 288 | . . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ ∃𝑓 ∈ (𝐴 ↑m ℕ)𝑓(⇝𝑡‘𝐽)𝑥)) |
17 | vex 3477 | . . . . 5 ⊢ 𝑥 ∈ V | |
18 | 17 | elima 6065 | . . . 4 ⊢ (𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝐴 ↑m ℕ)) ↔ ∃𝑓 ∈ (𝐴 ↑m ℕ)𝑓(⇝𝑡‘𝐽)𝑥) |
19 | 16, 18 | bitr4di 288 | . . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝐴 ↑m ℕ)))) |
20 | 19 | eqrdv 2729 | . 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) = ((⇝𝑡‘𝐽) “ (𝐴 ↑m ℕ))) |
21 | ovex 7445 | . . 3 ⊢ (𝐴 ↑m ℕ) ∈ V | |
22 | lmfun 23106 | . . . 4 ⊢ (𝐽 ∈ Haus → Fun (⇝𝑡‘𝐽)) | |
23 | 22 | 3ad2ant1 1132 | . . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → Fun (⇝𝑡‘𝐽)) |
24 | imadomg 10532 | . . 3 ⊢ ((𝐴 ↑m ℕ) ∈ V → (Fun (⇝𝑡‘𝐽) → ((⇝𝑡‘𝐽) “ (𝐴 ↑m ℕ)) ≼ (𝐴 ↑m ℕ))) | |
25 | 21, 23, 24 | mpsyl 68 | . 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ((⇝𝑡‘𝐽) “ (𝐴 ↑m ℕ)) ≼ (𝐴 ↑m ℕ)) |
26 | 20, 25 | eqbrtrd 5171 | 1 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ (𝐴 ↑m ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1780 ∈ wcel 2105 ∃wrex 3069 Vcvv 3473 ⊆ wss 3949 ∪ cuni 4909 class class class wbr 5149 “ cima 5680 Fun wfun 6538 ⟶wf 6540 ‘cfv 6544 (class class class)co 7412 ↑m cmap 8823 ≼ cdom 8940 ℕcn 12217 clsccl 22743 ⇝𝑡clm 22951 Hauscha 23033 1stωc1stc 23162 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-cc 10433 ax-ac2 10461 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-card 9937 df-acn 9940 df-ac 10114 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-top 22617 df-topon 22634 df-cld 22744 df-ntr 22745 df-cls 22746 df-lm 22954 df-haus 23040 df-1stc 23164 |
This theorem is referenced by: hauspwdom 23226 |
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