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Mirrors > Home > MPE Home > Th. List > kgentop | Structured version Visualization version GIF version |
Description: A compactly generated space is a topology. (Note: henceforth we will use the idiom "𝐽 ∈ ran 𝑘Gen " to denote "𝐽 is compactly generated", since as we will show a space is compactly generated iff it is in the range of the compact generator.) (Contributed by Mario Carneiro, 20-Mar-2015.) |
Ref | Expression |
---|---|
kgentop | ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kgenf 21869 | . . 3 ⊢ 𝑘Gen:Top⟶Top | |
2 | frn 6348 | . . 3 ⊢ (𝑘Gen:Top⟶Top → ran 𝑘Gen ⊆ Top) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ran 𝑘Gen ⊆ Top |
4 | 3 | sseli 3849 | 1 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2051 ⊆ wss 3824 ran crn 5405 ⟶wf 6182 Topctop 21221 𝑘Genckgen 21861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-oadd 7908 df-er 8088 df-en 8306 df-fin 8309 df-fi 8669 df-rest 16551 df-topgen 16572 df-top 21222 df-topon 21239 df-bases 21274 df-cmp 21715 df-kgen 21862 |
This theorem is referenced by: kgenidm 21875 iskgen2 21876 kgencn3 21886 txkgen 21980 qtopkgen 22038 |
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