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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 7467 | . . . . . 6 ⊢ ∪ 𝑗 ∈ V | |
2 | 1 | pwex 5283 | . . . . 5 ⊢ 𝒫 ∪ 𝑗 ∈ V |
3 | 2 | rabex 5237 | . . . 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 22144 | . . . 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 22150 | . . . 4 ⊢ (𝑥 ∈ Top → (𝑘Gen‘𝑥) ∈ Top) | |
8 | 7 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ Top) → (𝑘Gen‘𝑥) ∈ Top) |
9 | 4, 6, 8 | fmpt2d 6889 | . 2 ⊢ (⊤ → 𝑘Gen:Top⟶Top) |
10 | 9 | mptru 1544 | 1 ⊢ 𝑘Gen:Top⟶Top |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ∀wral 3140 {crab 3144 Vcvv 3496 ∩ cin 3937 𝒫 cpw 4541 ∪ cuni 4840 ↦ cmpt 5148 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↾t crest 16696 Topctop 21503 Compccmp 21996 𝑘Genckgen 22143 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-oadd 8108 df-er 8291 df-en 8512 df-fin 8515 df-fi 8877 df-rest 16698 df-topgen 16719 df-top 21504 df-topon 21521 df-bases 21556 df-cmp 21997 df-kgen 22144 |
This theorem is referenced by: kgentop 22152 kgenidm 22157 iskgen2 22158 kgen2cn 22169 |
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