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Mirrors > Home > MPE Home > Th. List > cnima | Structured version Visualization version GIF version |
Description: An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.) |
Ref | Expression |
---|---|
cnima | ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝐾) → (◡𝐹 “ 𝐴) ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2725 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | iscn2 23186 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))) |
4 | 3 | simprbi 495 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)) |
5 | 4 | simprd 494 | . 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽) |
6 | imaeq2 6060 | . . . 4 ⊢ (𝑥 = 𝐴 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝐴)) | |
7 | 6 | eleq1d 2810 | . . 3 ⊢ (𝑥 = 𝐴 → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ (◡𝐹 “ 𝐴) ∈ 𝐽)) |
8 | 7 | rspccva 3605 | . 2 ⊢ ((∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽 ∧ 𝐴 ∈ 𝐾) → (◡𝐹 “ 𝐴) ∈ 𝐽) |
9 | 5, 8 | sylan 578 | 1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝐾) → (◡𝐹 “ 𝐴) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∪ cuni 4909 ◡ccnv 5677 “ cima 5681 ⟶wf 6545 (class class class)co 7419 Topctop 22839 Cn ccn 23172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 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-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-map 8847 df-top 22840 df-topon 22857 df-cn 23175 |
This theorem is referenced by: cnco 23214 cnclima 23216 cnntri 23219 cnss1 23224 cnss2 23225 cncnpi 23226 cnrest 23233 cnt0 23294 cnhaus 23302 cncmp 23340 cnconn 23370 2ndcomap 23406 kgencn3 23506 txcnmpt 23572 txdis1cn 23583 pthaus 23586 ptrescn 23587 txkgen 23600 xkoco2cn 23606 xkococnlem 23607 txconn 23637 imasnopn 23638 qtopkgen 23658 qtopss 23663 isr0 23685 kqreglem1 23689 kqreglem2 23690 kqnrmlem1 23691 kqnrmlem2 23692 hmeoima 23713 hmeoopn 23714 hmeoimaf1o 23718 reghmph 23741 nrmhmph 23742 tmdgsum2 24044 symgtgp 24054 ghmcnp 24063 tgpt0 24067 qustgpopn 24068 qustgplem 24069 nmhmcn 25091 mbfimaopnlem 25628 cncombf 25631 cnmbf 25632 dvloglem 26627 efopnlem2 26636 efopn 26637 atansopn 26909 cnmbfm 34014 cvmsss2 35015 cvmliftmolem2 35023 cvmliftlem15 35039 cvmlift2lem9a 35044 cvmlift2lem9 35052 cvmlift2lem10 35053 cvmlift3lem6 35065 cvmlift3lem8 35067 dvtanlem 37273 rfcnpre1 44523 rfcnpre2 44535 icccncfext 45413 |
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