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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 7759 | . . . . . 6 ⊢ ∪ 𝑗 ∈ V | |
| 2 | 1 | pwex 5380 | . . . . 5 ⊢ 𝒫 ∪ 𝑗 ∈ V |
| 3 | 2 | rabex 5339 | . . . 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 23542 | . . . 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 23548 | . . . 4 ⊢ (𝑥 ∈ Top → (𝑘Gen‘𝑥) ∈ Top) | |
| 8 | 7 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ Top) → (𝑘Gen‘𝑥) ∈ Top) |
| 9 | 4, 6, 8 | fmpt2d 7144 | . 2 ⊢ (⊤ → 𝑘Gen:Top⟶Top) |
| 10 | 9 | mptru 1547 | 1 ⊢ 𝑘Gen:Top⟶Top |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ∩ cin 3950 𝒫 cpw 4600 ∪ cuni 4907 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↾t crest 17465 Topctop 22899 Compccmp 23394 𝑘Genckgen 23541 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-en 8986 df-fin 8989 df-fi 9451 df-rest 17467 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 df-cmp 23395 df-kgen 23542 |
| This theorem is referenced by: kgentop 23550 kgenidm 23555 iskgen2 23556 kgen2cn 23567 |
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