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Theorem qtopkgen 23213
Description: A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.)
Hypothesis
Ref Expression
qtopcmp.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
qtopkgen ((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ ran π‘˜Gen)
Proof of Theorem qtopkgen
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 kgentop 23045 . . 3 (𝐽 ∈ ran π‘˜Gen β†’ 𝐽 ∈ Top)
2 qtopcmp.1 . . . 4 𝑋 = βˆͺ 𝐽
32qtoptop 23203 . . 3 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ Top)
41, 3sylan 580 . 2 ((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ Top)
5 elssuni 4941 . . . . . . . 8 (π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹)) β†’ π‘₯ βŠ† βˆͺ (π‘˜Genβ€˜(𝐽 qTop 𝐹)))
65adantl 482 . . . . . . 7 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ π‘₯ βŠ† βˆͺ (π‘˜Genβ€˜(𝐽 qTop 𝐹)))
74adantr 481 . . . . . . . 8 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ (𝐽 qTop 𝐹) ∈ Top)
8 eqid 2732 . . . . . . . . 9 βˆͺ (𝐽 qTop 𝐹) = βˆͺ (𝐽 qTop 𝐹)
98kgenuni 23042 . . . . . . . 8 ((𝐽 qTop 𝐹) ∈ Top β†’ βˆͺ (𝐽 qTop 𝐹) = βˆͺ (π‘˜Genβ€˜(𝐽 qTop 𝐹)))
107, 9syl 17 . . . . . . 7 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ βˆͺ (𝐽 qTop 𝐹) = βˆͺ (π‘˜Genβ€˜(𝐽 qTop 𝐹)))
116, 10sseqtrrd 4023 . . . . . 6 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ π‘₯ βŠ† βˆͺ (𝐽 qTop 𝐹))
12 simpll 765 . . . . . . . 8 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ 𝐽 ∈ ran π‘˜Gen)
1312, 1syl 17 . . . . . . 7 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ 𝐽 ∈ Top)
14 simplr 767 . . . . . . . 8 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ 𝐹 Fn 𝑋)
15 dffn4 6811 . . . . . . . 8 (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–ontoβ†’ran 𝐹)
1614, 15sylib 217 . . . . . . 7 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ 𝐹:𝑋–ontoβ†’ran 𝐹)
172qtopuni 23205 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
1813, 16, 17syl2anc 584 . . . . . 6 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
1911, 18sseqtrrd 4023 . . . . 5 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ π‘₯ βŠ† ran 𝐹)
202toptopon 22418 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
2113, 20sylib 217 . . . . . . . 8 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
22 qtopid 23208 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
2321, 14, 22syl2anc 584 . . . . . . 7 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
24 kgencn3 23061 . . . . . . . 8 ((𝐽 ∈ ran π‘˜Gen ∧ (𝐽 qTop 𝐹) ∈ Top) β†’ (𝐽 Cn (𝐽 qTop 𝐹)) = (𝐽 Cn (π‘˜Genβ€˜(𝐽 qTop 𝐹))))
2512, 7, 24syl2anc 584 . . . . . . 7 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ (𝐽 Cn (𝐽 qTop 𝐹)) = (𝐽 Cn (π‘˜Genβ€˜(𝐽 qTop 𝐹))))
2623, 25eleqtrd 2835 . . . . . 6 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ 𝐹 ∈ (𝐽 Cn (π‘˜Genβ€˜(𝐽 qTop 𝐹))))
27 cnima 22768 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (π‘˜Genβ€˜(𝐽 qTop 𝐹))) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)
2826, 27sylancom 588 . . . . 5 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)
292elqtop2 23204 . . . . . 6 ((𝐽 ∈ ran π‘˜Gen ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ (π‘₯ ∈ (𝐽 qTop 𝐹) ↔ (π‘₯ βŠ† ran 𝐹 ∧ (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
3012, 16, 29syl2anc 584 . . . . 5 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ (π‘₯ ∈ (𝐽 qTop 𝐹) ↔ (π‘₯ βŠ† ran 𝐹 ∧ (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
3119, 28, 30mpbir2and 711 . . . 4 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ π‘₯ ∈ (𝐽 qTop 𝐹))
3231ex 413 . . 3 ((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) β†’ (π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹)) β†’ π‘₯ ∈ (𝐽 qTop 𝐹)))
3332ssrdv 3988 . 2 ((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) β†’ (π‘˜Genβ€˜(𝐽 qTop 𝐹)) βŠ† (𝐽 qTop 𝐹))
34 iskgen2 23051 . 2 ((𝐽 qTop 𝐹) ∈ ran π‘˜Gen ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ (π‘˜Genβ€˜(𝐽 qTop 𝐹)) βŠ† (𝐽 qTop 𝐹)))
354, 33, 34sylanbrc 583 1 ((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ ran π‘˜Gen)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  βˆͺ cuni 4908  β—‘ccnv 5675  ran crn 5677   β€œ cima 5679   Fn wfn 6538  β€“ontoβ†’wfo 6541  β€˜cfv 6543  (class class class)co 7408   qTop cqtop 17448  Topctop 22394  TopOnctopon 22411   Cn ccn 22727  π‘˜Genckgen 23036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-fin 8942  df-fi 9405  df-rest 17367  df-topgen 17388  df-qtop 17452  df-top 22395  df-topon 22412  df-bases 22448  df-cn 22730  df-cmp 22890  df-kgen 23037
This theorem is referenced by: (None)
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