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Theorem qtopkgen 23658
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 23490 . . 3 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
2 qtopcmp.1 . . . 4 𝑋 = 𝐽
32qtoptop 23648 . . 3 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
41, 3sylan 578 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
5 elssuni 4941 . . . . . . . 8 (𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹)) → 𝑥 (𝑘Gen‘(𝐽 qTop 𝐹)))
65adantl 480 . . . . . . 7 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 (𝑘Gen‘(𝐽 qTop 𝐹)))
74adantr 479 . . . . . . . 8 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ Top)
8 eqid 2725 . . . . . . . . 9 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
98kgenuni 23487 . . . . . . . 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, 10sseqtrrd 4018 . . . . . 6 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 (𝐽 qTop 𝐹))
12 simpll 765 . . . . . . . 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 767 . . . . . . . 8 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 Fn 𝑋)
15 dffn4 6816 . . . . . . . 8 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
1614, 15sylib 217 . . . . . . 7 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹:𝑋onto→ran 𝐹)
172qtopuni 23650 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐹:𝑋onto→ran 𝐹) → ran 𝐹 = (𝐽 qTop 𝐹))
1813, 16, 17syl2anc 582 . . . . . 6 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → ran 𝐹 = (𝐽 qTop 𝐹))
1911, 18sseqtrrd 4018 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ⊆ ran 𝐹)
202toptopon 22863 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
2113, 20sylib 217 . . . . . . . 8 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐽 ∈ (TopOn‘𝑋))
22 qtopid 23653 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
2321, 14, 22syl2anc 582 . . . . . . 7 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
24 kgencn3 23506 . . . . . . . 8 ((𝐽 ∈ ran 𝑘Gen ∧ (𝐽 qTop 𝐹) ∈ Top) → (𝐽 Cn (𝐽 qTop 𝐹)) = (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))))
2512, 7, 24syl2anc 582 . . . . . . 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 2827 . . . . . 6 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 ∈ (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))))
27 cnima 23213 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝐹𝑥) ∈ 𝐽)
2826, 27sylancom 586 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝐹𝑥) ∈ 𝐽)
292elqtop2 23649 . . . . . 6 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹:𝑋onto→ran 𝐹) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ ran 𝐹 ∧ (𝐹𝑥) ∈ 𝐽)))
3012, 16, 29syl2anc 582 . . . . 5 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ ran 𝐹 ∧ (𝐹𝑥) ∈ 𝐽)))
3119, 28, 30mpbir2and 711 . . . 4 (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ∈ (𝐽 qTop 𝐹))
3231ex 411 . . 3 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹)) → 𝑥 ∈ (𝐽 qTop 𝐹)))
3332ssrdv 3982 . 2 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝑘Gen‘(𝐽 qTop 𝐹)) ⊆ (𝐽 qTop 𝐹))
34 iskgen2 23496 . 2 ((𝐽 qTop 𝐹) ∈ ran 𝑘Gen ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ (𝑘Gen‘(𝐽 qTop 𝐹)) ⊆ (𝐽 qTop 𝐹)))
354, 33, 34sylanbrc 581 1 ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wss 3944   cuni 4909  ccnv 5677  ran crn 5679  cima 5681   Fn wfn 6544  ontowfo 6547  cfv 6549  (class class class)co 7419   qTop cqtop 17488  Topctop 22839  TopOnctopon 22856   Cn ccn 23172  𝑘Genckgen 23481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-fin 8968  df-fi 9436  df-rest 17407  df-topgen 17428  df-qtop 17492  df-top 22840  df-topon 22857  df-bases 22893  df-cn 23175  df-cmp 23335  df-kgen 23482
This theorem is referenced by: (None)
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