Theorem kqcld 21538

Description: The topological indistinguishability map is a closed map. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	|- F = ( x e. X |-> { y e. J | x e. y } )

Assertion

Ref	Expression
kqcld	|- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( F " U ) e. ( Clsd ` (KQ ` J ) ) )

Distinct variable groups:   x, y, J   x, X, y

Allowed substitution hints:   U( x, y)   F( x, y)

Proof of Theorem kqcld

Step	Hyp	Ref	Expression
1		imassrn 5477	. . . 4 |- ( F " U ) C_ ran F
2	1	a1i 11	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( F " U ) C_ ran F )
3		kqval.2	. . . . 5 |- F = ( x e. X |-> { y e. J | x e. y } )
4	3	kqcldsat 21536	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) = U )
5		simpr 477	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U e. ( Clsd ` J ) )
6	4, 5	eqeltrd 2701	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) e. ( Clsd ` J ) )
7	3	kqffn 21528	. . . . . 6 |- ( J e. (TopOn ` X ) -> F Fn X )
8		dffn4 6121	. . . . . 6 |- ( F Fn X <-> F : X -onto-> ran F )
9	7, 8	sylib 208	. . . . 5 |- ( J e. (TopOn ` X ) -> F : X -onto-> ran F )
10		qtopcld 21516	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ F : X -onto-> ran F ) -> ( ( F " U ) e. ( Clsd ` ( J qTop F ) ) <-> ( ( F " U ) C_ ran F /\ ( `' F " ( F " U ) ) e. ( Clsd ` J ) ) ) )
11	9, 10	mpdan 702	. . . 4 |- ( J e. (TopOn ` X ) -> ( ( F " U ) e. ( Clsd ` ( J qTop F ) ) <-> ( ( F " U ) C_ ran F /\ ( `' F " ( F " U ) ) e. ( Clsd ` J ) ) ) )
12	11	adantr 481	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( ( F " U ) e. ( Clsd ` ( J qTop F ) ) <-> ( ( F " U ) C_ ran F /\ ( `' F " ( F " U ) ) e. ( Clsd ` J ) ) ) )
13	2, 6, 12	mpbir2and 957	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( F " U ) e. ( Clsd ` ( J qTop F ) ) )
14	3	kqval 21529	. . . 4 |- ( J e. (TopOn ` X ) -> (KQ ` J ) = ( J qTop F ) )
15	14	adantr 481	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> (KQ ` J ) = ( J qTop F ) )
16	15	fveq2d 6195	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( Clsd ` (KQ ` J ) ) = ( Clsd ` ( J qTop F ) ) )
17	13, 16	eleqtrrd 2704	1 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( F " U ) e. ( Clsd ` (KQ ` J ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   |-> cmpt 4729   `'ccnv 5113   ran crn 5115   "cima 5117   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   qTop cqtop 16163  TopOnctopon 20715   Clsdccld 20820  KQckq 21496

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-kq 21497

This theorem is referenced by:  kqreglem1  21544  kqnrmlem1  21546  kqnrmlem2  21547