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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 2728 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2728 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | iscn2 23141 | . . 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 2099 ∀wral 3058 ∪ cuni 4908 ◡ccnv 5677 “ cima 5681 ⟶wf 6544 (class class class)co 7420 Topctop 22794 Cn ccn 23127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 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 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8846 df-top 22795 df-topon 22812 df-cn 23130 |
This theorem is referenced by: cnco 23169 cncls2i 23173 cnntri 23174 cnss1 23179 cncnpi 23181 cncnp2 23184 cnrest 23188 cnrest2r 23190 paste 23197 cncmp 23295 rncmp 23299 cnconn 23325 connima 23328 conncn 23329 2ndcomap 23361 kgen2cn 23462 txcnmpt 23527 uptx 23528 lmcn2 23552 xkoco1cn 23560 xkoco2cn 23561 xkococnlem 23562 cnmpt11 23566 cnmpt11f 23567 cnmpt1t 23568 cnmpt12 23570 cnmpt21 23574 cnmpt2t 23576 cnmpt22 23577 cnmpt22f 23578 cnmptcom 23581 cnmpt2k 23591 qtopeu 23619 hmeofval 23661 hmeof1o 23667 hmeontr 23672 hmeores 23674 hmeoqtop 23678 hmphen 23688 reghmph 23696 nrmhmph 23697 txhmeo 23706 xpstopnlem1 23712 flfcntr 23946 cnmpopc 24848 ishtpy 24897 htpyco1 24903 htpyco2 24904 isphtpy 24906 phtpyco2 24915 isphtpc 24919 pcofval 24936 pcopt 24948 pcopt2 24949 pcorevlem 24952 pi1cof 24985 pi1coghm 24987 cnmbfm 33883 cnpconn 34840 cnneiima 47935 |
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