Theorem kqsat 21534

Description: Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 21520). (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
kqsat	|- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( `' F " ( F " U ) ) = U )

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

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

Proof of Theorem kqsat

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . . 7 |- F = ( x e. X |-> { y e. J | x e. y } )
2	1	kqffn 21528	. . . . . 6 |- ( J e. (TopOn ` X ) -> F Fn X )
3		elpreima 6337	. . . . . 6 |- ( F Fn X -> ( z e. ( `' F " ( F " U ) ) <-> ( z e. X /\ ( F ` z ) e. ( F " U ) ) ) )
4	2, 3	syl 17	. . . . 5 |- ( J e. (TopOn ` X ) -> ( z e. ( `' F " ( F " U ) ) <-> ( z e. X /\ ( F ` z ) e. ( F " U ) ) ) )
5	4	adantr 481	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( z e. ( `' F " ( F " U ) ) <-> ( z e. X /\ ( F ` z ) e. ( F " U ) ) ) )
6	1	kqfvima 21533	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ U e. J /\ z e. X ) -> ( z e. U <-> ( F ` z ) e. ( F " U ) ) )
7	6	3expa 1265	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ z e. X ) -> ( z e. U <-> ( F ` z ) e. ( F " U ) ) )
8	7	biimprd 238	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ z e. X ) -> ( ( F ` z ) e. ( F " U ) -> z e. U ) )
9	8	expimpd 629	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( z e. X /\ ( F ` z ) e. ( F " U ) ) -> z e. U ) )
10	5, 9	sylbid 230	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( z e. ( `' F " ( F " U ) ) -> z e. U ) )
11	10	ssrdv 3609	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( `' F " ( F " U ) ) C_ U )
12		toponss 20731	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> U C_ X )
13		fndm 5990	. . . . . . 7 |- ( F Fn X -> dom F = X )
14	2, 13	syl 17	. . . . . 6 |- ( J e. (TopOn ` X ) -> dom F = X )
15	14	adantr 481	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> dom F = X )
16	12, 15	sseqtr4d 3642	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> U C_ dom F )
17		sseqin2 3817	. . . 4 |- ( U C_ dom F <-> ( dom F i^i U ) = U )
18	16, 17	sylib 208	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( dom F i^i U ) = U )
19		dminss 5547	. . 3 |- ( dom F i^i U ) C_ ( `' F " ( F " U ) )
20	18, 19	syl6eqssr 3656	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> U C_ ( `' F " ( F " U ) ) )
21	11, 20	eqssd 3620	1 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( `' F " ( F " U ) ) = U )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   i^i cin 3573   C_ wss 3574   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   ` cfv 5888  TopOnctopon 20715

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-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-rab 2921  df-v 3202  df-sbc 3436  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-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-fv 5896  df-topon 20716

This theorem is referenced by:  kqopn  21537  kqreglem2  21545  kqnrmlem2  21547