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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 2733 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2733 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | iscn2 22724 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))) |
4 | 3 | simplbi 499 | . 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
5 | 4 | simpld 496 | 1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ∀wral 3062 ∪ cuni 4907 ◡ccnv 5674 “ cima 5678 ⟶wf 6536 (class class class)co 7404 Topctop 22377 Cn ccn 22710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8818 df-top 22378 df-topon 22395 df-cn 22713 |
This theorem is referenced by: cnco 22752 cnclima 22754 cnntri 22757 cnclsi 22758 cnss2 22763 cncnpi 22764 cncnp2 22767 cnrest 22771 cnrest2 22772 cnrest2r 22773 lmcn 22791 cnt0 22832 cnt1 22836 cnhaus 22840 kgen2cn 23045 txcnmpt 23110 uptx 23111 txcn 23112 xkoco1cn 23143 xkoco2cn 23144 xkococnlem 23145 cnmpt21f 23158 qtopss 23201 qtopomap 23204 qtopcmap 23205 hmeofval 23244 hmeof1o 23250 hmeores 23257 hmphen 23271 txhmeo 23289 htpyco2 24477 hauseqcn 32816 cnmbfm 33200 hausgraph 41887 rfcnpre1 43636 fcnre 43642 cnneiima 47451 |
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