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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 23366 | . . 3 ⊢ 𝑘Gen:Top⟶Top | |
2 | frn 6724 | . . 3 ⊢ (𝑘Gen:Top⟶Top → ran 𝑘Gen ⊆ Top) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ran 𝑘Gen ⊆ Top |
4 | 3 | sseli 3978 | 1 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3948 ran crn 5677 ⟶wf 6539 Topctop 22716 𝑘Genckgen 23358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-en 8946 df-fin 8949 df-fi 9412 df-rest 17375 df-topgen 17396 df-top 22717 df-topon 22734 df-bases 22770 df-cmp 23212 df-kgen 23359 |
This theorem is referenced by: kgenidm 23372 iskgen2 23373 kgencn3 23383 txkgen 23477 qtopkgen 23535 |
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