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Theorem kgentop 22242
 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 22241 . . 3 𝑘Gen:Top⟶Top
2 frn 6504 . . 3 (𝑘Gen:Top⟶Top → ran 𝑘Gen ⊆ Top)
31, 2ax-mp 5 . 2 ran 𝑘Gen ⊆ Top
43sseli 3888 1 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111   ⊆ wss 3858  ran crn 5525  ⟶wf 6331  Topctop 21593  𝑘Genckgen 22233 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-en 8528  df-fin 8531  df-fi 8908  df-rest 16754  df-topgen 16775  df-top 21594  df-topon 21611  df-bases 21646  df-cmp 22087  df-kgen 22234 This theorem is referenced by:  kgenidm  22247  iskgen2  22248  kgencn3  22258  txkgen  22352  qtopkgen  22410
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