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Theorem qtopkgen 21453
Description: A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.)
Hypothesis
Ref Expression
qtopcmp.1 𝑋 = 𝐽
Assertion
Ref Expression
qtopkgen ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ ran 𝑘Gen)

Proof of Theorem qtopkgen
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 kgentop 21285 . . 3 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
2 qtopcmp.1 . . . 4 𝑋 = 𝐽
32qtoptop 21443 . . 3 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
41, 3sylan 488 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
5 elssuni 4440 . . . . . . . 8 (𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹)) → 𝑥 (𝑘Gen‘(𝐽 qTop 𝐹)))
65adantl 482 . . . . . . 7 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 (𝑘Gen‘(𝐽 qTop 𝐹)))
74adantr 481 . . . . . . . 8 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ Top)
8 eqid 2621 . . . . . . . . 9 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
98kgenuni 21282 . . . . . . . 8 ((𝐽 qTop 𝐹) ∈ Top → (𝐽 qTop 𝐹) = (𝑘Gen‘(𝐽 qTop 𝐹)))
107, 9syl 17 . . . . . . 7 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) = (𝑘Gen‘(𝐽 qTop 𝐹)))
116, 10sseqtr4d 3627 . . . . . 6 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 (𝐽 qTop 𝐹))
12 simpll 789 . . . . . . . 8 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐽 ∈ ran 𝑘Gen)
1312, 1syl 17 . . . . . . 7 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐽 ∈ Top)
14 simplr 791 . . . . . . . 8 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 Fn 𝑋)
15 dffn4 6088 . . . . . . . 8 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
1614, 15sylib 208 . . . . . . 7 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹:𝑋onto→ran 𝐹)
172qtopuni 21445 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐹:𝑋onto→ran 𝐹) → ran 𝐹 = (𝐽 qTop 𝐹))
1813, 16, 17syl2anc 692 . . . . . 6 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → ran 𝐹 = (𝐽 qTop 𝐹))
1911, 18sseqtr4d 3627 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ⊆ ran 𝐹)
202toptopon 20662 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
2113, 20sylib 208 . . . . . . . 8 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐽 ∈ (TopOn‘𝑋))
22 qtopid 21448 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
2321, 14, 22syl2anc 692 . . . . . . 7 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
24 kgencn3 21301 . . . . . . . 8 ((𝐽 ∈ ran 𝑘Gen ∧ (𝐽 qTop 𝐹) ∈ Top) → (𝐽 Cn (𝐽 qTop 𝐹)) = (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))))
2512, 7, 24syl2anc 692 . . . . . . 7 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝐽 Cn (𝐽 qTop 𝐹)) = (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))))
2623, 25eleqtrd 2700 . . . . . 6 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 ∈ (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))))
27 cnima 21009 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝐹𝑥) ∈ 𝐽)
2826, 27sylancom 700 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝐹𝑥) ∈ 𝐽)
292elqtop2 21444 . . . . . 6 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹:𝑋onto→ran 𝐹) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ ran 𝐹 ∧ (𝐹𝑥) ∈ 𝐽)))
3012, 16, 29syl2anc 692 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ ran 𝐹 ∧ (𝐹𝑥) ∈ 𝐽)))
3119, 28, 30mpbir2and 956 . . . 4 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ∈ (𝐽 qTop 𝐹))
3231ex 450 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹)) → 𝑥 ∈ (𝐽 qTop 𝐹)))
3332ssrdv 3594 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝑘Gen‘(𝐽 qTop 𝐹)) ⊆ (𝐽 qTop 𝐹))
34 iskgen2 21291 . 2 ((𝐽 qTop 𝐹) ∈ ran 𝑘Gen ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ (𝑘Gen‘(𝐽 qTop 𝐹)) ⊆ (𝐽 qTop 𝐹)))
354, 33, 34sylanbrc 697 1 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wss 3560   cuni 4409  ccnv 5083  ran crn 5085  cima 5087   Fn wfn 5852  ontowfo 5855  cfv 5857  (class class class)co 6615   qTop cqtop 16103  Topctop 20638  TopOnctopon 20655   Cn ccn 20968  𝑘Genckgen 21276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-fin 7919  df-fi 8277  df-rest 16023  df-topgen 16044  df-qtop 16107  df-top 20639  df-topon 20656  df-bases 20690  df-cn 20971  df-cmp 21130  df-kgen 21277
This theorem is referenced by: (None)
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