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Mirrors > Home > MPE Home > Th. List > imacmp | Structured version Visualization version GIF version |
Description: The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 18-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
imacmp | ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Comp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5691 | . . 3 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
2 | 1 | oveq2i 7431 | . 2 ⊢ (𝐾 ↾t (𝐹 “ 𝐴)) = (𝐾 ↾t ran (𝐹 ↾ 𝐴)) |
3 | simpr 484 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) ∈ Comp) | |
4 | simpl 482 | . . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
5 | inss2 4230 | . . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽 | |
6 | eqid 2728 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
7 | 6 | cnrest 23202 | . . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾)) |
8 | 4, 5, 7 | sylancl 585 | . . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾)) |
9 | resdmres 6236 | . . . . 5 ⊢ (𝐹 ↾ dom (𝐹 ↾ 𝐴)) = (𝐹 ↾ 𝐴) | |
10 | dmres 6007 | . . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹) | |
11 | eqid 2728 | . . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
12 | 6, 11 | cnf 23163 | . . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
13 | fdm 6731 | . . . . . . . . 9 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → dom 𝐹 = ∪ 𝐽) | |
14 | 4, 12, 13 | 3syl 18 | . . . . . . . 8 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → dom 𝐹 = ∪ 𝐽) |
15 | 14 | ineq2d 4212 | . . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐴 ∩ dom 𝐹) = (𝐴 ∩ ∪ 𝐽)) |
16 | 10, 15 | eqtrid 2780 | . . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → dom (𝐹 ↾ 𝐴) = (𝐴 ∩ ∪ 𝐽)) |
17 | 16 | reseq2d 5985 | . . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ dom (𝐹 ↾ 𝐴)) = (𝐹 ↾ (𝐴 ∩ ∪ 𝐽))) |
18 | 9, 17 | eqtr3id 2782 | . . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ 𝐴) = (𝐹 ↾ (𝐴 ∩ ∪ 𝐽))) |
19 | cmptop 23312 | . . . . . . 7 ⊢ ((𝐽 ↾t 𝐴) ∈ Comp → (𝐽 ↾t 𝐴) ∈ Top) | |
20 | 19 | adantl 481 | . . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) ∈ Top) |
21 | restrcl 23074 | . . . . . 6 ⊢ ((𝐽 ↾t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V)) | |
22 | 6 | restin 23083 | . . . . . 6 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽))) |
23 | 20, 21, 22 | 3syl 18 | . . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽))) |
24 | 23 | oveq1d 7435 | . . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → ((𝐽 ↾t 𝐴) Cn 𝐾) = ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾)) |
25 | 8, 18, 24 | 3eltr4d 2844 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) |
26 | rncmp 23313 | . . 3 ⊢ (((𝐽 ↾t 𝐴) ∈ Comp ∧ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) → (𝐾 ↾t ran (𝐹 ↾ 𝐴)) ∈ Comp) | |
27 | 3, 25, 26 | syl2anc 583 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t ran (𝐹 ↾ 𝐴)) ∈ Comp) |
28 | 2, 27 | eqeltrid 2833 | 1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∩ cin 3946 ⊆ wss 3947 ∪ cuni 4908 dom cdm 5678 ran crn 5679 ↾ cres 5680 “ cima 5681 ⟶wf 6544 (class class class)co 7420 ↾t crest 17402 Topctop 22808 Cn ccn 23141 Compccmp 23303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-fin 8968 df-fi 9435 df-rest 17404 df-topgen 17425 df-top 22809 df-topon 22826 df-bases 22862 df-cn 23144 df-cmp 23304 |
This theorem is referenced by: kgencn3 23475 txkgen 23569 xkoco1cn 23574 xkococnlem 23576 cmphaushmeo 23717 cnheiborlem 24893 |
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