Theorem qtopkgen 21513

Description: A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.)

Hypothesis

Ref	Expression
qtopcmp.1	|- X = U. J

Assertion

Ref	Expression
qtopkgen	|- ( ( J e. ran 𝑘Gen /\ F Fn X ) -> ( J qTop F ) e. ran 𝑘Gen )

Proof of Theorem qtopkgen

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kgentop 21345	. . 3 |- ( J e. ran 𝑘Gen -> J e. Top )
2		qtopcmp.1	. . . 4 |- X = U. J
3	2	qtoptop 21503	. . 3 |- ( ( J e. Top /\ F Fn X ) -> ( J qTop F ) e. Top )
4	1, 3	sylan 488	. 2 |- ( ( J e. ran 𝑘Gen /\ F Fn X ) -> ( J qTop F ) e. Top )
5		elssuni 4467	. . . . . . . 8 |- ( x e. (𝑘Gen ` ( J qTop F ) ) -> x C_ U. (𝑘Gen ` ( J qTop F ) ) )
6	5	adantl 482	. . . . . . 7 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> x C_ U. (𝑘Gen ` ( J qTop F ) ) )
7	4	adantr 481	. . . . . . . 8 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> ( J qTop F ) e. Top )
8		eqid 2622	. . . . . . . . 9 |- U. ( J qTop F ) = U. ( J qTop F )
9	8	kgenuni 21342	. . . . . . . 8 |- ( ( J qTop F ) e. Top -> U. ( J qTop F ) = U. (𝑘Gen ` ( J qTop F ) ) )
10	7, 9	syl 17	. . . . . . 7 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> U. ( J qTop F ) = U. (𝑘Gen ` ( J qTop F ) ) )
11	6, 10	sseqtr4d 3642	. . . . . 6 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> x C_ U. ( J qTop F ) )
12		simpll 790	. . . . . . . 8 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> J e. ran 𝑘Gen )
13	12, 1	syl 17	. . . . . . 7 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> J e. Top )
14		simplr 792	. . . . . . . 8 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> F Fn X )
15		dffn4 6121	. . . . . . . 8 |- ( F Fn X <-> F : X -onto-> ran F )
16	14, 15	sylib 208	. . . . . . 7 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> F : X -onto-> ran F )
17	2	qtopuni 21505	. . . . . . 7 |- ( ( J e. Top /\ F : X -onto-> ran F ) -> ran F = U. ( J qTop F ) )
18	13, 16, 17	syl2anc 693	. . . . . 6 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> ran F = U. ( J qTop F ) )
19	11, 18	sseqtr4d 3642	. . . . 5 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> x C_ ran F )
20	2	toptopon 20722	. . . . . . . . 9 |- ( J e. Top <-> J e. (TopOn ` X ) )
21	13, 20	sylib 208	. . . . . . . 8 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> J e. (TopOn ` X ) )
22		qtopid 21508	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ F Fn X ) -> F e. ( J Cn ( J qTop F ) ) )
23	21, 14, 22	syl2anc 693	. . . . . . 7 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> F e. ( J Cn ( J qTop F ) ) )
24		kgencn3 21361	. . . . . . . 8 |- ( ( J e. ran 𝑘Gen /\ ( J qTop F ) e. Top ) -> ( J Cn ( J qTop F ) ) = ( J Cn (𝑘Gen ` ( J qTop F ) ) ) )
25	12, 7, 24	syl2anc 693	. . . . . . 7 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> ( J Cn ( J qTop F ) ) = ( J Cn (𝑘Gen ` ( J qTop F ) ) ) )
26	23, 25	eleqtrd 2703	. . . . . 6 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> F e. ( J Cn (𝑘Gen ` ( J qTop F ) ) ) )
27		cnima 21069	. . . . . 6 |- ( ( F e. ( J Cn (𝑘Gen ` ( J qTop F ) ) ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> ( `' F " x ) e. J )
28	26, 27	sylancom 701	. . . . 5 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> ( `' F " x ) e. J )
29	2	elqtop2 21504	. . . . . 6 |- ( ( J e. ran 𝑘Gen /\ F : X -onto-> ran F ) -> ( x e. ( J qTop F ) <-> ( x C_ ran F /\ ( `' F " x ) e. J ) ) )
30	12, 16, 29	syl2anc 693	. . . . 5 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> ( x e. ( J qTop F ) <-> ( x C_ ran F /\ ( `' F " x ) e. J ) ) )
31	19, 28, 30	mpbir2and 957	. . . 4 |- ( ( ( J e. ran 𝑘Gen /\ F Fn X ) /\ x e. (𝑘Gen ` ( J qTop F ) ) ) -> x e. ( J qTop F ) )
32	31	ex 450	. . 3 |- ( ( J e. ran 𝑘Gen /\ F Fn X ) -> ( x e. (𝑘Gen ` ( J qTop F ) ) -> x e. ( J qTop F ) ) )
33	32	ssrdv 3609	. 2 |- ( ( J e. ran 𝑘Gen /\ F Fn X ) -> (𝑘Gen ` ( J qTop F ) ) C_ ( J qTop F ) )
34		iskgen2 21351	. 2 |- ( ( J qTop F ) e. ran 𝑘Gen <-> ( ( J qTop F ) e. Top /\ (𝑘Gen ` ( J qTop F ) ) C_ ( J qTop F ) ) )
35	4, 33, 34	sylanbrc 698	1 |- ( ( J e. ran 𝑘Gen /\ F Fn X ) -> ( J qTop F ) e. ran 𝑘Gen )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   U.cuni 4436   `'ccnv 5113   ran crn 5115   "cima 5117   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   qTop cqtop 16163   Topctop 20698  TopOnctopon 20715   Cn ccn 21028  𝑘Genckgen 21336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-qtop 16167  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cmp 21190  df-kgen 21337

This theorem is referenced by: (None)