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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 2610 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2610 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | iscn2 20852 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))) |
4 | 3 | simplbi 475 | . 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
5 | 4 | simprd 478 | 1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 ∪ cuni 4372 ◡ccnv 5037 “ cima 5041 ⟶wf 5800 (class class class)co 6549 Topctop 20517 Cn ccn 20838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-top 20521 df-topon 20523 df-cn 20841 |
This theorem is referenced by: cnco 20880 cncls2i 20884 cnntri 20885 cnss1 20890 cncnpi 20892 cncnp2 20895 cnrest 20899 cnrest2r 20901 paste 20908 cncmp 21005 rncmp 21009 cnconn 21035 conima 21038 concn 21039 2ndcomap 21071 kgen2cn 21172 txcnmpt 21237 uptx 21238 lmcn2 21262 xkoco1cn 21270 xkoco2cn 21271 xkococnlem 21272 cnmpt11 21276 cnmpt11f 21277 cnmpt1t 21278 cnmpt12 21280 cnmpt21 21284 cnmpt2t 21286 cnmpt22 21287 cnmpt22f 21288 cnmptcom 21291 cnmpt2k 21301 qtopeu 21329 hmeofval 21371 hmeof1o 21377 hmeontr 21382 hmeores 21384 hmeoqtop 21388 hmphen 21398 reghmph 21406 nrmhmph 21407 txhmeo 21416 xpstopnlem1 21422 flfcntr 21657 cnmpt2pc 22535 ishtpy 22579 htpyco1 22585 htpyco2 22586 isphtpy 22588 phtpyco2 22597 isphtpc 22601 pcofval 22618 pcopt 22630 pcopt2 22631 pcorevlem 22634 pi1cof 22667 pi1coghm 22669 cnmbfm 29652 cnpcon 30466 |
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