| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 2737 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 2 | eqid 2737 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
| 3 | 1, 2 | iscn2 23246 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))) |
| 4 | 3 | simprbi 496 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)) |
| 5 | 4 | simprd 495 | . 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽) |
| 6 | imaeq2 6074 | . . . 4 ⊢ (𝑥 = 𝐴 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝐴)) | |
| 7 | 6 | eleq1d 2826 | . . 3 ⊢ (𝑥 = 𝐴 → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ (◡𝐹 “ 𝐴) ∈ 𝐽)) |
| 8 | 7 | rspccva 3621 | . 2 ⊢ ((∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽 ∧ 𝐴 ∈ 𝐾) → (◡𝐹 “ 𝐴) ∈ 𝐽) |
| 9 | 5, 8 | sylan 580 | 1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝐾) → (◡𝐹 “ 𝐴) ∈ 𝐽) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∪ cuni 4907 ◡ccnv 5684 “ cima 5688 ⟶wf 6557 (class class class)co 7431 Topctop 22899 Cn ccn 23232 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-top 22900 df-topon 22917 df-cn 23235 |
| This theorem is referenced by: cnco 23274 cnclima 23276 cnntri 23279 cnss1 23284 cnss2 23285 cncnpi 23286 cnrest 23293 cnt0 23354 cnhaus 23362 cncmp 23400 cnconn 23430 2ndcomap 23466 kgencn3 23566 txcnmpt 23632 txdis1cn 23643 pthaus 23646 ptrescn 23647 txkgen 23660 xkoco2cn 23666 xkococnlem 23667 txconn 23697 imasnopn 23698 qtopkgen 23718 qtopss 23723 isr0 23745 kqreglem1 23749 kqreglem2 23750 kqnrmlem1 23751 kqnrmlem2 23752 hmeoima 23773 hmeoopn 23774 hmeoimaf1o 23778 reghmph 23801 nrmhmph 23802 tmdgsum2 24104 symgtgp 24114 ghmcnp 24123 tgpt0 24127 qustgpopn 24128 qustgplem 24129 nmhmcn 25153 mbfimaopnlem 25690 cncombf 25693 cnmbf 25694 dvloglem 26690 efopnlem2 26699 efopn 26700 atansopn 26975 cnmbfm 34265 cvmsss2 35279 cvmliftmolem2 35287 cvmliftlem15 35303 cvmlift2lem9a 35308 cvmlift2lem9 35316 cvmlift2lem10 35317 cvmlift3lem6 35329 cvmlift3lem8 35331 dvtanlem 37676 resuppsinopn 42393 rfcnpre1 45024 rfcnpre2 45036 icccncfext 45902 |
| Copyright terms: Public domain | W3C validator |