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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 2738 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2738 | . . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | cnf 21997 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
4 | 3 | adantr 484 | . . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
5 | ffun 6507 | . . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Fun 𝐹) | |
6 | funcnvcnv 6406 | . . . . . 6 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
7 | imadif 6423 | . . . . . 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 6526 | . . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽) | |
10 | 9 | difeq1d 4012 | . . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → ((◡𝐹 “ ∪ 𝐾) ∖ (◡𝐹 “ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴))) |
11 | 8, 10 | eqtr2d 2774 | . . . 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 21782 | . . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐾) → (∪ 𝐾 ∖ 𝐴) ∈ 𝐾) |
14 | cnima 22016 | . . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (∪ 𝐾 ∖ 𝐴) ∈ 𝐾) → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) ∈ 𝐽) | |
15 | 13, 14 | sylan2 596 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) ∈ 𝐽) |
16 | 12, 15 | eqeltrd 2833 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽) |
17 | cntop1 21991 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
18 | cnvimass 5923 | . . . 4 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 | |
19 | 18, 4 | fssdm 6524 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽) |
20 | 1 | iscld2 21779 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽)) |
21 | 17, 19, 20 | syl2an2r 685 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽)) |
22 | 16, 21 | mpbird 260 | 1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∖ cdif 3840 ⊆ wss 3843 ∪ cuni 4796 ◡ccnv 5524 “ cima 5528 Fun wfun 6333 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 Topctop 21644 Clsdccld 21767 Cn ccn 21975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-map 8439 df-top 21645 df-topon 21662 df-cld 21770 df-cn 21978 |
This theorem is referenced by: iscncl 22020 cncls2i 22021 paste 22045 cnt1 22101 dnsconst 22129 cnconn 22173 hauseqlcld 22397 txconn 22440 imasncld 22442 r0cld 22489 kqreglem2 22493 kqnrmlem1 22494 kqnrmlem2 22495 hmeocld 22518 nrmhmph 22545 tgphaus 22868 csscld 24001 clsocv 24002 hmeoclda 34160 hmeocldb 34161 rfcnpre3 42114 rfcnpre4 42115 sepfsepc 45743 seppcld 45745 |
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