Theorem qtopss 21518

Description: A surjective continuous function from J to K induces a topology J qTop F on the base set of K. This topology is in general finer than K. Together with qtopid 21508, this implies that J qTop F is the finest topology making F continuous, i.e. the final topology with respect to the family { F }. (Contributed by Mario Carneiro, 24-Mar-2015.)

Assertion

Ref	Expression
qtopss	|- ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) -> K C_ ( J qTop F ) )

Proof of Theorem qtopss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		toponss 20731	. . . . 5 |- ( ( K e. (TopOn ` Y ) /\ x e. K ) -> x C_ Y )
2	1	3ad2antl2 1224	. . . 4 |- ( ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) /\ x e. K ) -> x C_ Y )
3		cnima 21069	. . . . 5 |- ( ( F e. ( J Cn K ) /\ x e. K ) -> ( `' F " x ) e. J )
4	3	3ad2antl1 1223	. . . 4 |- ( ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) /\ x e. K ) -> ( `' F " x ) e. J )
5		simpl1 1064	. . . . . . 7 |- ( ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) /\ x e. K ) -> F e. ( J Cn K ) )
6		cntop1 21044	. . . . . . 7 |- ( F e. ( J Cn K ) -> J e. Top )
7	5, 6	syl 17	. . . . . 6 |- ( ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) /\ x e. K ) -> J e. Top )
8		eqid 2622	. . . . . . 7 |- U. J = U. J
9	8	toptopon 20722	. . . . . 6 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
10	7, 9	sylib 208	. . . . 5 |- ( ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) /\ x e. K ) -> J e. (TopOn ` U. J ) )
11		simpl2 1065	. . . . . . . 8 |- ( ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) /\ x e. K ) -> K e. (TopOn ` Y ) )
12		cnf2 21053	. . . . . . . 8 |- ( ( J e. (TopOn ` U. J ) /\ K e. (TopOn ` Y ) /\ F e. ( J Cn K ) ) -> F : U. J --> Y )
13	10, 11, 5, 12	syl3anc 1326	. . . . . . 7 |- ( ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) /\ x e. K ) -> F : U. J --> Y )
14		ffn 6045	. . . . . . 7 |- ( F : U. J --> Y -> F Fn U. J )
15	13, 14	syl 17	. . . . . 6 |- ( ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) /\ x e. K ) -> F Fn U. J )
16		simpl3 1066	. . . . . 6 |- ( ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) /\ x e. K ) -> ran F = Y )
17		df-fo 5894	. . . . . 6 |- ( F : U. J -onto-> Y <-> ( F Fn U. J /\ ran F = Y ) )
18	15, 16, 17	sylanbrc 698	. . . . 5 |- ( ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) /\ x e. K ) -> F : U. J -onto-> Y )
19		elqtop3 21506	. . . . 5 |- ( ( J e. (TopOn ` U. J ) /\ F : U. J -onto-> Y ) -> ( x e. ( J qTop F ) <-> ( x C_ Y /\ ( `' F " x ) e. J ) ) )
20	10, 18, 19	syl2anc 693	. . . 4 |- ( ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) /\ x e. K ) -> ( x e. ( J qTop F ) <-> ( x C_ Y /\ ( `' F " x ) e. J ) ) )
21	2, 4, 20	mpbir2and 957	. . 3 |- ( ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) /\ x e. K ) -> x e. ( J qTop F ) )
22	21	ex 450	. 2 |- ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) -> ( x e. K -> x e. ( J qTop F ) ) )
23	22	ssrdv 3609	1 |- ( ( F e. ( J Cn K ) /\ K e. (TopOn ` Y ) /\ ran F = Y ) -> K C_ ( J qTop F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   U.cuni 4436   `'ccnv 5113   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   qTop cqtop 16163   Topctop 20698  TopOnctopon 20715   Cn ccn 21028

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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-map 7859  df-qtop 16167  df-top 20699  df-topon 20716  df-cn 21031

This theorem is referenced by:  qtoprest  21520  qtopomap  21521  qtopcmap  21522