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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 2818 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2818 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | iscn2 21774 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))) |
4 | 3 | simplbi 498 | . 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
5 | 4 | simprd 496 | 1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ∀wral 3135 ∪ cuni 4830 ◡ccnv 5547 “ cima 5551 ⟶wf 6344 (class class class)co 7145 Topctop 21429 Cn ccn 21760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-top 21430 df-topon 21447 df-cn 21763 |
This theorem is referenced by: cnco 21802 cncls2i 21806 cnntri 21807 cnss1 21812 cncnpi 21814 cncnp2 21817 cnrest 21821 cnrest2r 21823 paste 21830 cncmp 21928 rncmp 21932 cnconn 21958 connima 21961 conncn 21962 2ndcomap 21994 kgen2cn 22095 txcnmpt 22160 uptx 22161 lmcn2 22185 xkoco1cn 22193 xkoco2cn 22194 xkococnlem 22195 cnmpt11 22199 cnmpt11f 22200 cnmpt1t 22201 cnmpt12 22203 cnmpt21 22207 cnmpt2t 22209 cnmpt22 22210 cnmpt22f 22211 cnmptcom 22214 cnmpt2k 22224 qtopeu 22252 hmeofval 22294 hmeof1o 22300 hmeontr 22305 hmeores 22307 hmeoqtop 22311 hmphen 22321 reghmph 22329 nrmhmph 22330 txhmeo 22339 xpstopnlem1 22345 flfcntr 22579 cnmpopc 23459 ishtpy 23503 htpyco1 23509 htpyco2 23510 isphtpy 23512 phtpyco2 23521 isphtpc 23525 pcofval 23541 pcopt 23553 pcopt2 23554 pcorevlem 23557 pi1cof 23590 pi1coghm 23592 cnmbfm 31420 cnpconn 32374 |
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