Theorem qtopid 21508

Description: A quotient map is a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)

Assertion

Ref	Expression
qtopid	|- ( ( J e. (TopOn ` X ) /\ F Fn X ) -> F e. ( J Cn ( J qTop F ) ) )

Proof of Theorem qtopid

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( J e. (TopOn ` X ) /\ F Fn X ) -> F Fn X )
2		dffn4 6121	. . . 4 |- ( F Fn X <-> F : X -onto-> ran F )
3	1, 2	sylib 208	. . 3 |- ( ( J e. (TopOn ` X ) /\ F Fn X ) -> F : X -onto-> ran F )
4		fof 6115	. . 3 |- ( F : X -onto-> ran F -> F : X --> ran F )
5	3, 4	syl 17	. 2 |- ( ( J e. (TopOn ` X ) /\ F Fn X ) -> F : X --> ran F )
6		elqtop3 21506	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ F : X -onto-> ran F ) -> ( x e. ( J qTop F ) <-> ( x C_ ran F /\ ( `' F " x ) e. J ) ) )
7	3, 6	syldan 487	. . . 4 |- ( ( J e. (TopOn ` X ) /\ F Fn X ) -> ( x e. ( J qTop F ) <-> ( x C_ ran F /\ ( `' F " x ) e. J ) ) )
8	7	simplbda 654	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ F Fn X ) /\ x e. ( J qTop F ) ) -> ( `' F " x ) e. J )
9	8	ralrimiva 2966	. 2 |- ( ( J e. (TopOn ` X ) /\ F Fn X ) -> A. x e. ( J qTop F ) ( `' F " x ) e. J )
10		qtoptopon 21507	. . . 4 |- ( ( J e. (TopOn ` X ) /\ F : X -onto-> ran F ) -> ( J qTop F ) e. (TopOn ` ran F ) )
11	3, 10	syldan 487	. . 3 |- ( ( J e. (TopOn ` X ) /\ F Fn X ) -> ( J qTop F ) e. (TopOn ` ran F ) )
12		iscn 21039	. . 3 |- ( ( J e. (TopOn ` X ) /\ ( J qTop F ) e. (TopOn ` ran F ) ) -> ( F e. ( J Cn ( J qTop F ) ) <-> ( F : X --> ran F /\ A. x e. ( J qTop F ) ( `' F " x ) e. J ) ) )
13	11, 12	syldan 487	. 2 |- ( ( J e. (TopOn ` X ) /\ F Fn X ) -> ( F e. ( J Cn ( J qTop F ) ) <-> ( F : X --> ran F /\ A. x e. ( J qTop F ) ( `' F " x ) e. J ) ) )
14	5, 9, 13	mpbir2and 957	1 |- ( ( J e. (TopOn ` X ) /\ F Fn X ) -> F e. ( J Cn ( J qTop F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   C_ wss 3574   `'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  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:  qtopcmplem  21510  qtopkgen  21513  qtoprest  21520  kqid  21531  qtopf1  21619  qtophmeo  21620  qustgplem  21924  circcn  29905