Theorem qtopt1 29902

Description: If every equivalence class is closed, then the quotient space is T₁ . (Contributed by Thierry Arnoux, 5-Jan-2020.)

Hypotheses

Ref	Expression
qtopt1.x	|- X = U. J
qtopt1.1	|- ( ph -> J e. Fre )
qtopt1.2	|- ( ph -> F : X -onto-> Y )
qtopt1.3	|- ( ( ph /\ x e. Y ) -> ( `' F " { x } ) e. ( Clsd ` J ) )

Assertion

Ref	Expression
qtopt1	|- ( ph -> ( J qTop F ) e. Fre )

Distinct variable groups:   x, F   x, J   ph, x

Allowed substitution hints:   X( x)   Y( x)

Proof of Theorem qtopt1

Step	Hyp	Ref	Expression
1		qtopt1.1	. . . 4 |- ( ph -> J e. Fre )
2		t1top 21134	. . . 4 |- ( J e. Fre -> J e. Top )
3	1, 2	syl 17	. . 3 |- ( ph -> J e. Top )
4		qtopt1.2	. . . 4 |- ( ph -> F : X -onto-> Y )
5		fofn 6117	. . . 4 |- ( F : X -onto-> Y -> F Fn X )
6	4, 5	syl 17	. . 3 |- ( ph -> F Fn X )
7		qtopt1.x	. . . 4 |- X = U. J
8	7	qtoptop 21503	. . 3 |- ( ( J e. Top /\ F Fn X ) -> ( J qTop F ) e. Top )
9	3, 6, 8	syl2anc 693	. 2 |- ( ph -> ( J qTop F ) e. Top )
10		simpr 477	. . . . . 6 |- ( ( ph /\ x e. U. ( J qTop F ) ) -> x e. U. ( J qTop F ) )
11	7	qtopuni 21505	. . . . . . . 8 |- ( ( J e. Top /\ F : X -onto-> Y ) -> Y = U. ( J qTop F ) )
12	3, 4, 11	syl2anc 693	. . . . . . 7 |- ( ph -> Y = U. ( J qTop F ) )
13	12	adantr 481	. . . . . 6 |- ( ( ph /\ x e. U. ( J qTop F ) ) -> Y = U. ( J qTop F ) )
14	10, 13	eleqtrrd 2704	. . . . 5 |- ( ( ph /\ x e. U. ( J qTop F ) ) -> x e. Y )
15	14	snssd 4340	. . . 4 |- ( ( ph /\ x e. U. ( J qTop F ) ) -> { x } C_ Y )
16		qtopt1.3	. . . . 5 |- ( ( ph /\ x e. Y ) -> ( `' F " { x } ) e. ( Clsd ` J ) )
17	14, 16	syldan 487	. . . 4 |- ( ( ph /\ x e. U. ( J qTop F ) ) -> ( `' F " { x } ) e. ( Clsd ` J ) )
18	3, 7	jctir 561	. . . . . . 7 |- ( ph -> ( J e. Top /\ X = U. J ) )
19		istopon 20717	. . . . . . 7 |- ( J e. (TopOn ` X ) <-> ( J e. Top /\ X = U. J ) )
20	18, 19	sylibr 224	. . . . . 6 |- ( ph -> J e. (TopOn ` X ) )
21		qtopcld 21516	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ F : X -onto-> Y ) -> ( { x } e. ( Clsd ` ( J qTop F ) ) <-> ( { x } C_ Y /\ ( `' F " { x } ) e. ( Clsd ` J ) ) ) )
22	20, 4, 21	syl2anc 693	. . . . 5 |- ( ph -> ( { x } e. ( Clsd ` ( J qTop F ) ) <-> ( { x } C_ Y /\ ( `' F " { x } ) e. ( Clsd ` J ) ) ) )
23	22	adantr 481	. . . 4 |- ( ( ph /\ x e. U. ( J qTop F ) ) -> ( { x } e. ( Clsd ` ( J qTop F ) ) <-> ( { x } C_ Y /\ ( `' F " { x } ) e. ( Clsd ` J ) ) ) )
24	15, 17, 23	mpbir2and 957	. . 3 |- ( ( ph /\ x e. U. ( J qTop F ) ) -> { x } e. ( Clsd ` ( J qTop F ) ) )
25	24	ralrimiva 2966	. 2 |- ( ph -> A. x e. U. ( J qTop F ) { x } e. ( Clsd ` ( J qTop F ) ) )
26		eqid 2622	. . 3 |- U. ( J qTop F ) = U. ( J qTop F )
27	26	ist1 21125	. 2 |- ( ( J qTop F ) e. Fre <-> ( ( J qTop F ) e. Top /\ A. x e. U. ( J qTop F ) { x } e. ( Clsd ` ( J qTop F ) ) ) )
28	9, 25, 27	sylanbrc 698	1 |- ( ph -> ( J qTop F ) e. Fre )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   {csn 4177   U.cuni 4436   `'ccnv 5113   "cima 5117   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   qTop cqtop 16163   Topctop 20698  TopOnctopon 20715   Clsdccld 20820   Frect1 21111

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-qtop 16167  df-top 20699  df-topon 20716  df-cld 20823  df-t1 21118

This theorem is referenced by: (None)