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| Mirrors > Home > MPE Home > Th. List > cntop1 | Structured version Visualization version GIF version | ||
| Description: Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.) |
| Ref | Expression |
|---|---|
| cntop1 | ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ 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 | simpld 494 | 1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ 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 cnclima 23276 cnntri 23279 cnclsi 23280 cnss2 23285 cncnpi 23286 cncnp2 23289 cnrest 23293 cnrest2 23294 cnrest2r 23295 lmcn 23313 cnt0 23354 cnt1 23358 cnhaus 23362 kgen2cn 23567 txcnmpt 23632 uptx 23633 txcn 23634 xkoco1cn 23665 xkoco2cn 23666 xkococnlem 23667 cnmpt21f 23680 qtopss 23723 qtopomap 23726 qtopcmap 23727 hmeofval 23766 hmeof1o 23772 hmeores 23779 hmphen 23793 txhmeo 23811 htpyco2 25011 hauseqcn 33897 cnmbfm 34265 hausgraph 43217 rfcnpre1 45024 fcnre 45030 cnneiima 48814 |
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