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 2610 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2610 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | iscn2 20852 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))) |
4 | 3 | simprbi 479 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)) |
5 | 4 | simprd 478 | . 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽) |
6 | imaeq2 5381 | . . . 4 ⊢ (𝑥 = 𝐴 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝐴)) | |
7 | 6 | eleq1d 2672 | . . 3 ⊢ (𝑥 = 𝐴 → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ (◡𝐹 “ 𝐴) ∈ 𝐽)) |
8 | 7 | rspccva 3281 | . 2 ⊢ ((∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽 ∧ 𝐴 ∈ 𝐾) → (◡𝐹 “ 𝐴) ∈ 𝐽) |
9 | 5, 8 | sylan 487 | 1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝐾) → (◡𝐹 “ 𝐴) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∪ cuni 4372 ◡ccnv 5037 “ cima 5041 ⟶wf 5800 (class class class)co 6549 Topctop 20517 Cn ccn 20838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-top 20521 df-topon 20523 df-cn 20841 |
This theorem is referenced by: cnco 20880 cnclima 20882 cnntri 20885 cnss1 20890 cnss2 20891 cncnpi 20892 cnrest 20899 cnt0 20960 cnhaus 20968 cncmp 21005 cnconn 21035 2ndcomap 21071 kgencn3 21171 txcnmpt 21237 txdis1cn 21248 pthaus 21251 ptrescn 21252 txkgen 21265 xkoco2cn 21271 xkococnlem 21272 txcon 21302 imasnopn 21303 qtopkgen 21323 qtopss 21328 isr0 21350 kqreglem1 21354 kqreglem2 21355 kqnrmlem1 21356 kqnrmlem2 21357 hmeoima 21378 hmeoopn 21379 hmeoimaf1o 21383 reghmph 21406 nrmhmph 21407 tmdgsum2 21710 symgtgp 21715 ghmcnp 21728 tgpt0 21732 qustgpopn 21733 qustgplem 21734 nmhmcn 22728 mbfimaopnlem 23228 cncombf 23231 cnmbf 23232 dvloglem 24194 efopnlem2 24203 efopn 24204 atansopn 24459 cnmbfm 29652 cvmsss2 30510 cvmliftmolem2 30518 cvmliftlem15 30534 cvmlift2lem9a 30539 cvmlift2lem9 30547 cvmlift2lem10 30548 cvmlift3lem6 30560 cvmlift3lem8 30562 dvtanlem 32629 rfcnpre1 38201 rfcnpre2 38213 icccncfext 38773 |
Copyright terms: Public domain | W3C validator |