Theorem kqdisj 21535

Description: A version of imain 5974 for the topological indistinguishability 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
kqdisj	|- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( F " U ) i^i ( F " ( A \ U ) ) ) = (/) )

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

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

Proof of Theorem kqdisj

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imadmres 5627	. . . . 5 |- ( F " dom ( F |` ( A \ U ) ) ) = ( F " ( A \ U ) )
2		dmres 5419	. . . . . . 7 |- dom ( F |` ( A \ U ) ) = ( ( A \ U ) i^i dom F )
3		kqval.2	. . . . . . . . . . 11 |- F = ( x e. X |-> { y e. J | x e. y } )
4	3	kqffn 21528	. . . . . . . . . 10 |- ( J e. (TopOn ` X ) -> F Fn X )
5	4	adantr 481	. . . . . . . . 9 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> F Fn X )
6		fndm 5990	. . . . . . . . 9 |- ( F Fn X -> dom F = X )
7	5, 6	syl 17	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> dom F = X )
8	7	ineq2d 3814	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( A \ U ) i^i dom F ) = ( ( A \ U ) i^i X ) )
9	2, 8	syl5eq 2668	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> dom ( F |` ( A \ U ) ) = ( ( A \ U ) i^i X ) )
10	9	imaeq2d 5466	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " dom ( F |` ( A \ U ) ) ) = ( F " ( ( A \ U ) i^i X ) ) )
11	1, 10	syl5eqr 2670	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " ( A \ U ) ) = ( F " ( ( A \ U ) i^i X ) ) )
12		indif1 3871	. . . . . 6 |- ( ( A \ U ) i^i X ) = ( ( A i^i X ) \ U )
13		inss2 3834	. . . . . . 7 |- ( A i^i X ) C_ X
14		ssdif 3745	. . . . . . 7 |- ( ( A i^i X ) C_ X -> ( ( A i^i X ) \ U ) C_ ( X \ U ) )
15	13, 14	ax-mp 5	. . . . . 6 |- ( ( A i^i X ) \ U ) C_ ( X \ U )
16	12, 15	eqsstri 3635	. . . . 5 |- ( ( A \ U ) i^i X ) C_ ( X \ U )
17		imass2 5501	. . . . 5 |- ( ( ( A \ U ) i^i X ) C_ ( X \ U ) -> ( F " ( ( A \ U ) i^i X ) ) C_ ( F " ( X \ U ) ) )
18	16, 17	mp1i 13	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " ( ( A \ U ) i^i X ) ) C_ ( F " ( X \ U ) ) )
19	11, 18	eqsstrd 3639	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " ( A \ U ) ) C_ ( F " ( X \ U ) ) )
20		sslin 3839	. . 3 |- ( ( F " ( A \ U ) ) C_ ( F " ( X \ U ) ) -> ( ( F " U ) i^i ( F " ( A \ U ) ) ) C_ ( ( F " U ) i^i ( F " ( X \ U ) ) ) )
21	19, 20	syl 17	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( F " U ) i^i ( F " ( A \ U ) ) ) C_ ( ( F " U ) i^i ( F " ( X \ U ) ) ) )
22		eldifn 3733	. . . . . . 7 |- ( w e. ( X \ U ) -> -. w e. U )
23	22	adantl 482	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> -. w e. U )
24		simpll 790	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> J e. (TopOn ` X ) )
25		simplr 792	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> U e. J )
26		eldifi 3732	. . . . . . . 8 |- ( w e. ( X \ U ) -> w e. X )
27	26	adantl 482	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> w e. X )
28	3	kqfvima 21533	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ U e. J /\ w e. X ) -> ( w e. U <-> ( F ` w ) e. ( F " U ) ) )
29	24, 25, 27, 28	syl3anc 1326	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> ( w e. U <-> ( F ` w ) e. ( F " U ) ) )
30	23, 29	mtbid 314	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> -. ( F ` w ) e. ( F " U ) )
31	30	ralrimiva 2966	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> A. w e. ( X \ U ) -. ( F ` w ) e. ( F " U ) )
32		difss 3737	. . . . 5 |- ( X \ U ) C_ X
33		eleq1 2689	. . . . . . 7 |- ( z = ( F ` w ) -> ( z e. ( F " U ) <-> ( F ` w ) e. ( F " U ) ) )
34	33	notbid 308	. . . . . 6 |- ( z = ( F ` w ) -> ( -. z e. ( F " U ) <-> -. ( F ` w ) e. ( F " U ) ) )
35	34	ralima 6498	. . . . 5 |- ( ( F Fn X /\ ( X \ U ) C_ X ) -> ( A. z e. ( F " ( X \ U ) ) -. z e. ( F " U ) <-> A. w e. ( X \ U ) -. ( F ` w ) e. ( F " U ) ) )
36	5, 32, 35	sylancl 694	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( A. z e. ( F " ( X \ U ) ) -. z e. ( F " U ) <-> A. w e. ( X \ U ) -. ( F ` w ) e. ( F " U ) ) )
37	31, 36	mpbird 247	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> A. z e. ( F " ( X \ U ) ) -. z e. ( F " U ) )
38		disjr 4018	. . 3 |- ( ( ( F " U ) i^i ( F " ( X \ U ) ) ) = (/) <-> A. z e. ( F " ( X \ U ) ) -. z e. ( F " U ) )
39	37, 38	sylibr 224	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( F " U ) i^i ( F " ( X \ U ) ) ) = (/) )
40		sseq0 3975	. 2 |- ( ( ( ( F " U ) i^i ( F " ( A \ U ) ) ) C_ ( ( F " U ) i^i ( F " ( X \ U ) ) ) /\ ( ( F " U ) i^i ( F " ( X \ U ) ) ) = (/) ) -> ( ( F " U ) i^i ( F " ( A \ U ) ) ) = (/) )
41	21, 39, 40	syl2anc 693	1 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( F " U ) i^i ( F " ( A \ U ) ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   dom cdm 5114   |` cres 5116   "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:  kqcldsat  21536  regr1lem  21542