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Theorem qtopkgen 23084
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 22916 . . 3 (𝐽 ∈ ran π‘˜Gen β†’ 𝐽 ∈ Top)
2 qtopcmp.1 . . . 4 𝑋 = βˆͺ 𝐽
32qtoptop 23074 . . 3 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ Top)
41, 3sylan 581 . 2 ((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ Top)
5 elssuni 4902 . . . . . . . 8 (π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹)) β†’ π‘₯ βŠ† βˆͺ (π‘˜Genβ€˜(𝐽 qTop 𝐹)))
65adantl 483 . . . . . . 7 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ π‘₯ βŠ† βˆͺ (π‘˜Genβ€˜(𝐽 qTop 𝐹)))
74adantr 482 . . . . . . . 8 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ (𝐽 qTop 𝐹) ∈ Top)
8 eqid 2733 . . . . . . . . 9 βˆͺ (𝐽 qTop 𝐹) = βˆͺ (𝐽 qTop 𝐹)
98kgenuni 22913 . . . . . . . 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 3989 . . . . . 6 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ π‘₯ βŠ† βˆͺ (𝐽 qTop 𝐹))
12 simpll 766 . . . . . . . 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 768 . . . . . . . 8 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ 𝐹 Fn 𝑋)
15 dffn4 6766 . . . . . . . 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 23076 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
1813, 16, 17syl2anc 585 . . . . . 6 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
1911, 18sseqtrrd 3989 . . . . 5 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ π‘₯ βŠ† ran 𝐹)
202toptopon 22289 . . . . . . . . 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 23079 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
2321, 14, 22syl2anc 585 . . . . . . 7 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
24 kgencn3 22932 . . . . . . . 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 585 . . . . . . 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 2836 . . . . . 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 22639 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (π‘˜Genβ€˜(𝐽 qTop 𝐹))) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)
2826, 27sylancom 589 . . . . 5 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)
292elqtop2 23075 . . . . . 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 585 . . . . 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 712 . . . 4 (((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹))) β†’ π‘₯ ∈ (𝐽 qTop 𝐹))
3231ex 414 . . 3 ((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) β†’ (π‘₯ ∈ (π‘˜Genβ€˜(𝐽 qTop 𝐹)) β†’ π‘₯ ∈ (𝐽 qTop 𝐹)))
3332ssrdv 3954 . 2 ((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) β†’ (π‘˜Genβ€˜(𝐽 qTop 𝐹)) βŠ† (𝐽 qTop 𝐹))
34 iskgen2 22922 . 2 ((𝐽 qTop 𝐹) ∈ ran π‘˜Gen ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ (π‘˜Genβ€˜(𝐽 qTop 𝐹)) βŠ† (𝐽 qTop 𝐹)))
354, 33, 34sylanbrc 584 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 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3914  βˆͺ cuni 4869  β—‘ccnv 5636  ran crn 5638   β€œ cima 5640   Fn wfn 6495  β€“ontoβ†’wfo 6498  β€˜cfv 6500  (class class class)co 7361   qTop cqtop 17393  Topctop 22265  TopOnctopon 22282   Cn ccn 22598  π‘˜Genckgen 22907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-qtop 17397  df-top 22266  df-topon 22283  df-bases 22319  df-cn 22601  df-cmp 22761  df-kgen 22908
This theorem is referenced by: (None)
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