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Mirrors > Home > MPE Home > Th. List > cnclima | Structured version Visualization version GIF version |
Description: A closed subset of the codomain of a continuous function has a closed preimage. (Contributed by NM, 15-Mar-2007.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnclima | ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2731 | . . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | cnf 23069 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
4 | 3 | adantr 480 | . . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
5 | ffun 6720 | . . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Fun 𝐹) | |
6 | funcnvcnv 6615 | . . . . . 6 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
7 | imadif 6632 | . . . . . 6 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) = ((◡𝐹 “ ∪ 𝐾) ∖ (◡𝐹 “ 𝐴))) | |
8 | 5, 6, 7 | 3syl 18 | . . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) = ((◡𝐹 “ ∪ 𝐾) ∖ (◡𝐹 “ 𝐴))) |
9 | fimacnv 6739 | . . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽) | |
10 | 9 | difeq1d 4121 | . . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → ((◡𝐹 “ ∪ 𝐾) ∖ (◡𝐹 “ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴))) |
11 | 8, 10 | eqtr2d 2772 | . . . 4 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) = (◡𝐹 “ (∪ 𝐾 ∖ 𝐴))) |
12 | 4, 11 | syl 17 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) = (◡𝐹 “ (∪ 𝐾 ∖ 𝐴))) |
13 | 2 | cldopn 22854 | . . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐾) → (∪ 𝐾 ∖ 𝐴) ∈ 𝐾) |
14 | cnima 23088 | . . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (∪ 𝐾 ∖ 𝐴) ∈ 𝐾) → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) ∈ 𝐽) | |
15 | 13, 14 | sylan2 592 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) ∈ 𝐽) |
16 | 12, 15 | eqeltrd 2832 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽) |
17 | cntop1 23063 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
18 | cnvimass 6080 | . . . 4 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 | |
19 | 18, 4 | fssdm 6737 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽) |
20 | 1 | iscld2 22851 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽)) |
21 | 17, 19, 20 | syl2an2r 682 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽)) |
22 | 16, 21 | mpbird 257 | 1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∖ cdif 3945 ⊆ wss 3948 ∪ cuni 4908 ◡ccnv 5675 “ cima 5679 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 Topctop 22714 Clsdccld 22839 Cn ccn 23047 |
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-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-map 8828 df-top 22715 df-topon 22732 df-cld 22842 df-cn 23050 |
This theorem is referenced by: iscncl 23092 cncls2i 23093 paste 23117 cnt1 23173 dnsconst 23201 cnconn 23245 hauseqlcld 23469 txconn 23512 imasncld 23514 r0cld 23561 kqreglem2 23565 kqnrmlem1 23566 kqnrmlem2 23567 hmeocld 23590 nrmhmph 23617 tgphaus 23940 csscld 25096 clsocv 25097 hmeoclda 35681 hmeocldb 35682 rfcnpre3 44179 rfcnpre4 44180 sepfsepc 47721 seppcld 47723 |
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