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Theorem kgentop 21870
 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.)
Assertion
Ref Expression
kgentop (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)

Proof of Theorem kgentop
StepHypRef Expression
1 kgenf 21869 . . 3 𝑘Gen:Top⟶Top
2 frn 6348 . . 3 (𝑘Gen:Top⟶Top → ran 𝑘Gen ⊆ Top)
31, 2ax-mp 5 . 2 ran 𝑘Gen ⊆ Top
43sseli 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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