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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 7733 | . . . . . 6 ⊢ ∪ 𝑗 ∈ V | |
2 | 1 | pwex 5378 | . . . . 5 ⊢ 𝒫 ∪ 𝑗 ∈ V |
3 | 2 | rabex 5332 | . . . 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 23358 | . . . 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 23364 | . . . 4 ⊢ (𝑥 ∈ Top → (𝑘Gen‘𝑥) ∈ Top) | |
8 | 7 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ Top) → (𝑘Gen‘𝑥) ∈ Top) |
9 | 4, 6, 8 | fmpt2d 7125 | . 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 2105 ∀wral 3060 {crab 3431 Vcvv 3473 ∩ cin 3947 𝒫 cpw 4602 ∪ cuni 4908 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ↾t crest 17373 Topctop 22715 Compccmp 23210 𝑘Genckgen 23357 |
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 22716 df-topon 22733 df-bases 22769 df-cmp 23211 df-kgen 23358 |
This theorem is referenced by: kgentop 23366 kgenidm 23371 iskgen2 23372 kgen2cn 23383 |
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