Theorem qtopf1 21619

Description: If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypotheses

Ref	Expression
qtopf1.1	|- ( ph -> J e. (TopOn ` X ) )
qtopf1.2	|- ( ph -> F : X -1-1-> Y )

Assertion

Ref	Expression
qtopf1	|- ( ph -> F e. ( J Homeo ( J qTop F ) ) )

Proof of Theorem qtopf1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qtopf1.1	. . 3 |- ( ph -> J e. (TopOn ` X ) )
2		qtopf1.2	. . . 4 |- ( ph -> F : X -1-1-> Y )
3		f1fn 6102	. . . 4 |- ( F : X -1-1-> Y -> F Fn X )
4	2, 3	syl 17	. . 3 |- ( ph -> F Fn X )
5		qtopid 21508	. . 3 |- ( ( J e. (TopOn ` X ) /\ F Fn X ) -> F e. ( J Cn ( J qTop F ) ) )
6	1, 4, 5	syl2anc 693	. 2 |- ( ph -> F e. ( J Cn ( J qTop F ) ) )
7		f1f1orn 6148	. . . 4 |- ( F : X -1-1-> Y -> F : X -1-1-onto-> ran F )
8		f1ocnv 6149	. . . 4 |- ( F : X -1-1-onto-> ran F -> `' F : ran F -1-1-onto-> X )
9		f1of 6137	. . . 4 |- ( `' F : ran F -1-1-onto-> X -> `' F : ran F --> X )
10	2, 7, 8, 9	4syl 19	. . 3 |- ( ph -> `' F : ran F --> X )
11		imacnvcnv 5599	. . . . 5 |- ( `' `' F " x ) = ( F " x )
12		imassrn 5477	. . . . . . 7 |- ( F " x ) C_ ran F
13	12	a1i 11	. . . . . 6 |- ( ( ph /\ x e. J ) -> ( F " x ) C_ ran F )
14	2	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. J ) -> F : X -1-1-> Y )
15		toponss 20731	. . . . . . . . 9 |- ( ( J e. (TopOn ` X ) /\ x e. J ) -> x C_ X )
16	1, 15	sylan 488	. . . . . . . 8 |- ( ( ph /\ x e. J ) -> x C_ X )
17		f1imacnv 6153	. . . . . . . 8 |- ( ( F : X -1-1-> Y /\ x C_ X ) -> ( `' F " ( F " x ) ) = x )
18	14, 16, 17	syl2anc 693	. . . . . . 7 |- ( ( ph /\ x e. J ) -> ( `' F " ( F " x ) ) = x )
19		simpr 477	. . . . . . 7 |- ( ( ph /\ x e. J ) -> x e. J )
20	18, 19	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ x e. J ) -> ( `' F " ( F " x ) ) e. J )
21		dffn4 6121	. . . . . . . . 9 |- ( F Fn X <-> F : X -onto-> ran F )
22	4, 21	sylib 208	. . . . . . . 8 |- ( ph -> F : X -onto-> ran F )
23		elqtop3 21506	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ F : X -onto-> ran F ) -> ( ( F " x ) e. ( J qTop F ) <-> ( ( F " x ) C_ ran F /\ ( `' F " ( F " x ) ) e. J ) ) )
24	1, 22, 23	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( F " x ) e. ( J qTop F ) <-> ( ( F " x ) C_ ran F /\ ( `' F " ( F " x ) ) e. J ) ) )
25	24	adantr 481	. . . . . 6 |- ( ( ph /\ x e. J ) -> ( ( F " x ) e. ( J qTop F ) <-> ( ( F " x ) C_ ran F /\ ( `' F " ( F " x ) ) e. J ) ) )
26	13, 20, 25	mpbir2and 957	. . . . 5 |- ( ( ph /\ x e. J ) -> ( F " x ) e. ( J qTop F ) )
27	11, 26	syl5eqel 2705	. . . 4 |- ( ( ph /\ x e. J ) -> ( `' `' F " x ) e. ( J qTop F ) )
28	27	ralrimiva 2966	. . 3 |- ( ph -> A. x e. J ( `' `' F " x ) e. ( J qTop F ) )
29		qtoptopon 21507	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ F : X -onto-> ran F ) -> ( J qTop F ) e. (TopOn ` ran F ) )
30	1, 22, 29	syl2anc 693	. . . 4 |- ( ph -> ( J qTop F ) e. (TopOn ` ran F ) )
31		iscn 21039	. . . 4 |- ( ( ( J qTop F ) e. (TopOn ` ran F ) /\ J e. (TopOn ` X ) ) -> ( `' F e. ( ( J qTop F ) Cn J ) <-> ( `' F : ran F --> X /\ A. x e. J ( `' `' F " x ) e. ( J qTop F ) ) ) )
32	30, 1, 31	syl2anc 693	. . 3 |- ( ph -> ( `' F e. ( ( J qTop F ) Cn J ) <-> ( `' F : ran F --> X /\ A. x e. J ( `' `' F " x ) e. ( J qTop F ) ) ) )
33	10, 28, 32	mpbir2and 957	. 2 |- ( ph -> `' F e. ( ( J qTop F ) Cn J ) )
34		ishmeo 21562	. 2 |- ( F e. ( J Homeo ( J qTop F ) ) <-> ( F e. ( J Cn ( J qTop F ) ) /\ `' F e. ( ( J qTop F ) Cn J ) ) )
35	6, 33, 34	sylanbrc 698	1 |- ( ph -> F e. ( J Homeo ( J qTop F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   `'ccnv 5113   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   qTop cqtop 16163  TopOnctopon 20715   Cn ccn 21028   Homeochmeo 21556

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  df-hmeo 21558

This theorem is referenced by:  t0kq  21621