Theorem kqdisj 20746

Description: A version of imain 5677 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 5346	. . . . 5 |- ( F " dom ( F |` ( A \ U ) ) ) = ( F " ( A \ U ) )
2		dmres 5144	. . . . . . 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 20739	. . . . . . . . . 10 |- ( J e. (TopOn ` X ) -> F Fn X )
5	4	adantr 466	. . . . . . . . 9 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> F Fn X )
6		fndm 5693	. . . . . . . . 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 3664	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( A \ U ) i^i dom F ) = ( ( A \ U ) i^i X ) )
9	2, 8	syl5eq 2475	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> dom ( F |` ( A \ U ) ) = ( ( A \ U ) i^i X ) )
10	9	imaeq2d 5187	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " dom ( F |` ( A \ U ) ) ) = ( F " ( ( A \ U ) i^i X ) ) )
11	1, 10	syl5eqr 2477	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " ( A \ U ) ) = ( F " ( ( A \ U ) i^i X ) ) )
12		indif1 3717	. . . . . 6 |- ( ( A \ U ) i^i X ) = ( ( A i^i X ) \ U )
13		inss2 3683	. . . . . . 7 |- ( A i^i X ) C_ X
14		ssdif 3600	. . . . . . 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 3494	. . . . 5 |- ( ( A \ U ) i^i X ) C_ ( X \ U )
17		imass2 5223	. . . . 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 3498	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " ( A \ U ) ) C_ ( F " ( X \ U ) ) )
20		sslin 3688	. . 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 3588	. . . . . . 7 |- ( w e. ( X \ U ) -> -. w e. U )
23	22	adantl 467	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> -. w e. U )
24		simpll 758	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> J e. (TopOn ` X ) )
25		simplr 760	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> U e. J )
26		eldifi 3587	. . . . . . . 8 |- ( w e. ( X \ U ) -> w e. X )
27	26	adantl 467	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> w e. X )
28	3	kqfvima 20744	. . . . . . 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 1264	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> ( w e. U <-> ( F ` w ) e. ( F " U ) ) )
30	23, 29	mtbid 301	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> -. ( F ` w ) e. ( F " U ) )
31	30	ralrimiva 2836	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> A. w e. ( X \ U ) -. ( F ` w ) e. ( F " U ) )
32		difss 3592	. . . . 5 |- ( X \ U ) C_ X
33		eleq1 2495	. . . . . . 7 |- ( z = ( F ` w ) -> ( z e. ( F " U ) <-> ( F ` w ) e. ( F " U ) ) )
34	33	notbid 295	. . . . . 6 |- ( z = ( F ` w ) -> ( -. z e. ( F " U ) <-> -. ( F ` w ) e. ( F " U ) ) )
35	34	ralima 6161	. . . . 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 666	. . . 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 235	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> A. z e. ( F " ( X \ U ) ) -. z e. ( F " U ) )
38		disjr 3836	. . 3 |- ( ( ( F " U ) i^i ( F " ( X \ U ) ) ) = (/) <-> A. z e. ( F " ( X \ U ) ) -. z e. ( F " U ) )
39	37, 38	sylibr 215	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( F " U ) i^i ( F " ( X \ U ) ) ) = (/) )
40		sseq0 3796	. 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 665	1 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( F " U ) i^i ( F " ( A \ U ) ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1872   A.wral 2771   {crab 2775   \ cdif 3433   i^i cin 3435   C_ wss 3436   (/)c0 3761   |-> cmpt 4482   dom cdm 4853   |` cres 4855   "cima 4856   Fn wfn 5596   ` cfv 5601  TopOnctopon 19917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-topon 19922

This theorem is referenced by:  kqcldsat  20747  regr1lem  20753