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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 2821 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2821 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | iscn2 21776 | . . 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 3138 ∪ cuni 4832 ◡ccnv 5548 “ cima 5552 ⟶wf 6345 (class class class)co 7145 Topctop 21431 Cn ccn 21762 |
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 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8398 df-top 21432 df-topon 21449 df-cn 21765 |
This theorem is referenced by: cnco 21804 cncls2i 21808 cnntri 21809 cnss1 21814 cncnpi 21816 cncnp2 21819 cnrest 21823 cnrest2r 21825 paste 21832 cncmp 21930 rncmp 21934 cnconn 21960 connima 21963 conncn 21964 2ndcomap 21996 kgen2cn 22097 txcnmpt 22162 uptx 22163 lmcn2 22187 xkoco1cn 22195 xkoco2cn 22196 xkococnlem 22197 cnmpt11 22201 cnmpt11f 22202 cnmpt1t 22203 cnmpt12 22205 cnmpt21 22209 cnmpt2t 22211 cnmpt22 22212 cnmpt22f 22213 cnmptcom 22216 cnmpt2k 22226 qtopeu 22254 hmeofval 22296 hmeof1o 22302 hmeontr 22307 hmeores 22309 hmeoqtop 22313 hmphen 22323 reghmph 22331 nrmhmph 22332 txhmeo 22341 xpstopnlem1 22347 flfcntr 22581 cnmpopc 23461 ishtpy 23505 htpyco1 23511 htpyco2 23512 isphtpy 23514 phtpyco2 23523 isphtpc 23527 pcofval 23543 pcopt 23555 pcopt2 23556 pcorevlem 23559 pi1cof 23592 pi1coghm 23594 cnmbfm 31421 cnpconn 32375 |
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