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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 2739 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2739 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | iscn2 22002 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))) |
4 | 3 | simplbi 501 | . 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
5 | 4 | simpld 498 | 1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2114 ∀wral 3054 ∪ cuni 4806 ◡ccnv 5534 “ cima 5538 ⟶wf 6346 (class class class)co 7183 Topctop 21657 Cn ccn 21988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7186 df-oprab 7187 df-mpo 7188 df-map 8452 df-top 21658 df-topon 21675 df-cn 21991 |
This theorem is referenced by: cnco 22030 cnclima 22032 cnntri 22035 cnclsi 22036 cnss2 22041 cncnpi 22042 cncnp2 22045 cnrest 22049 cnrest2 22050 cnrest2r 22051 lmcn 22069 cnt0 22110 cnt1 22114 cnhaus 22118 kgen2cn 22323 txcnmpt 22388 uptx 22389 txcn 22390 xkoco1cn 22421 xkoco2cn 22422 xkococnlem 22423 cnmpt21f 22436 qtopss 22479 qtopomap 22482 qtopcmap 22483 hmeofval 22522 hmeof1o 22528 hmeores 22535 hmphen 22549 txhmeo 22567 htpyco2 23744 hauseqcn 31433 cnmbfm 31813 hausgraph 40650 rfcnpre1 42141 fcnre 42147 cnneiima 45780 |
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