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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 21566 | . . 3 ⊢ 𝑘Gen:Top⟶Top | |
2 | frn 6194 | . . 3 ⊢ (𝑘Gen:Top⟶Top → ran 𝑘Gen ⊆ Top) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ran 𝑘Gen ⊆ Top |
4 | 3 | sseli 3749 | 1 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2145 ⊆ wss 3724 ran crn 5251 ⟶wf 6028 Topctop 20919 𝑘Genckgen 21558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-1st 7316 df-2nd 7317 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-oadd 7718 df-er 7897 df-en 8111 df-fin 8114 df-fi 8474 df-rest 16292 df-topgen 16313 df-top 20920 df-topon 20937 df-bases 20972 df-cmp 21412 df-kgen 21559 |
This theorem is referenced by: kgenidm 21572 iskgen2 21573 kgencn3 21583 txkgen 21677 qtopkgen 21735 |
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