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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 7315 | . . . . . 6 ⊢ ∪ 𝑗 ∈ V | |
2 | 1 | pwex 5165 | . . . . 5 ⊢ 𝒫 ∪ 𝑗 ∈ V |
3 | 2 | rabex 5119 | . . . 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 21814 | . . . 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 21820 | . . . 4 ⊢ (𝑥 ∈ Top → (𝑘Gen‘𝑥) ∈ Top) | |
8 | 7 | adantl 482 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ Top) → (𝑘Gen‘𝑥) ∈ Top) |
9 | 4, 6, 8 | fmpt2d 6741 | . 2 ⊢ (⊤ → 𝑘Gen:Top⟶Top) |
10 | 9 | mptru 1527 | 1 ⊢ 𝑘Gen:Top⟶Top |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ⊤wtru 1521 ∈ wcel 2079 ∀wral 3103 {crab 3107 Vcvv 3432 ∩ cin 3853 𝒫 cpw 4447 ∪ cuni 4739 ↦ cmpt 5035 ⟶wf 6213 ‘cfv 6217 (class class class)co 7007 ↾t crest 16511 Topctop 21173 Compccmp 21666 𝑘Genckgen 21813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-oadd 7948 df-er 8130 df-en 8348 df-fin 8351 df-fi 8711 df-rest 16513 df-topgen 16534 df-top 21174 df-topon 21191 df-bases 21226 df-cmp 21667 df-kgen 21814 |
This theorem is referenced by: kgentop 21822 kgenidm 21827 iskgen2 21828 kgen2cn 21839 |
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