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| Mirrors > Home > MPE Home > Th. List > cntop2 | Structured version Visualization version GIF version | ||
| Description: Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.) |
| Ref | Expression |
|---|---|
| cntop2 | ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 2 | eqid 2737 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
| 3 | 1, 2 | iscn2 23246 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))) |
| 4 | 3 | simplbi 497 | . 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
| 5 | 4 | simprd 495 | 1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ 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 cncls2i 23278 cnntri 23279 cnss1 23284 cncnpi 23286 cncnp2 23289 cnrest 23293 cnrest2r 23295 paste 23302 cncmp 23400 rncmp 23404 cnconn 23430 connima 23433 conncn 23434 2ndcomap 23466 kgen2cn 23567 txcnmpt 23632 uptx 23633 lmcn2 23657 xkoco1cn 23665 xkoco2cn 23666 xkococnlem 23667 cnmpt11 23671 cnmpt11f 23672 cnmpt1t 23673 cnmpt12 23675 cnmpt21 23679 cnmpt2t 23681 cnmpt22 23682 cnmpt22f 23683 cnmptcom 23686 cnmpt2k 23696 qtopeu 23724 hmeofval 23766 hmeof1o 23772 hmeontr 23777 hmeores 23779 hmeoqtop 23783 hmphen 23793 reghmph 23801 nrmhmph 23802 txhmeo 23811 xpstopnlem1 23817 flfcntr 24051 cnmpopc 24955 ishtpy 25004 htpyco1 25010 htpyco2 25011 isphtpy 25013 phtpyco2 25022 isphtpc 25026 pcofval 25043 pcopt 25055 pcopt2 25056 pcorevlem 25059 pi1cof 25092 pi1coghm 25094 cnmbfm 34265 cnpconn 35235 cnneiima 48814 |
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