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Mirrors > Home > MPE Home > Th. List > kgenf | Structured version Visualization version GIF version |
Description: The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.) |
Ref | Expression |
---|---|
kgenf | ⊢ 𝑘Gen:Top⟶Top |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vuniex 7757 | . . . . . 6 ⊢ ∪ 𝑗 ∈ V | |
2 | 1 | pwex 5385 | . . . . 5 ⊢ 𝒫 ∪ 𝑗 ∈ V |
3 | 2 | rabex 5344 | . . . 4 ⊢ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))} ∈ V |
4 | 3 | a1i 11 | . . 3 ⊢ ((⊤ ∧ 𝑗 ∈ Top) → {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))} ∈ V) |
5 | df-kgen 23557 | . . . 4 ⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))}) | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))})) |
7 | kgenftop 23563 | . . . 4 ⊢ (𝑥 ∈ Top → (𝑘Gen‘𝑥) ∈ Top) | |
8 | 7 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ Top) → (𝑘Gen‘𝑥) ∈ Top) |
9 | 4, 6, 8 | fmpt2d 7143 | . 2 ⊢ (⊤ → 𝑘Gen:Top⟶Top) |
10 | 9 | mptru 1543 | 1 ⊢ 𝑘Gen:Top⟶Top |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ⊤wtru 1537 ∈ wcel 2105 ∀wral 3058 {crab 3432 Vcvv 3477 ∩ cin 3961 𝒫 cpw 4604 ∪ cuni 4911 ↦ cmpt 5230 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ↾t crest 17466 Topctop 22914 Compccmp 23409 𝑘Genckgen 23556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-en 8984 df-fin 8987 df-fi 9448 df-rest 17468 df-topgen 17489 df-top 22915 df-topon 22932 df-bases 22968 df-cmp 23410 df-kgen 23557 |
This theorem is referenced by: kgentop 23565 kgenidm 23570 iskgen2 23571 kgen2cn 23582 |
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