Theorem kqdisj 19436

Description: A version of imain 5601 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 5437	. . . . 5 |- ( F " dom ( F |` ( A \ U ) ) ) = ( F " ( A \ U ) )
2		dmres 5238	. . . . . . 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 19429	. . . . . . . . . 10 |- ( J e. (TopOn ` X ) -> F Fn X )
5	4	adantr 465	. . . . . . . . 9 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> F Fn X )
6		fndm 5617	. . . . . . . . 9 |- ( F Fn X -> dom F = X )
7	5, 6	syl 16	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> dom F = X )
8	7	ineq2d 3659	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( A \ U ) i^i dom F ) = ( ( A \ U ) i^i X ) )
9	2, 8	syl5eq 2507	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> dom ( F |` ( A \ U ) ) = ( ( A \ U ) i^i X ) )
10	9	imaeq2d 5276	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " dom ( F |` ( A \ U ) ) ) = ( F " ( ( A \ U ) i^i X ) ) )
11	1, 10	syl5eqr 2509	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " ( A \ U ) ) = ( F " ( ( A \ U ) i^i X ) ) )
12		indif1 3701	. . . . . 6 |- ( ( A \ U ) i^i X ) = ( ( A i^i X ) \ U )
13		inss2 3678	. . . . . . 7 |- ( A i^i X ) C_ X
14		ssdif 3598	. . . . . . 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 3493	. . . . 5 |- ( ( A \ U ) i^i X ) C_ ( X \ U )
17		imass2 5311	. . . . 5 |- ( ( ( A \ U ) i^i X ) C_ ( X \ U ) -> ( F " ( ( A \ U ) i^i X ) ) C_ ( F " ( X \ U ) ) )
18	16, 17	mp1i 12	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " ( ( A \ U ) i^i X ) ) C_ ( F " ( X \ U ) ) )
19	11, 18	eqsstrd 3497	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " ( A \ U ) ) C_ ( F " ( X \ U ) ) )
20		sslin 3683	. . 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 16	. 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 3586	. . . . . . 7 |- ( w e. ( X \ U ) -> -. w e. U )
23	22	adantl 466	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> -. w e. U )
24		simpll 753	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> J e. (TopOn ` X ) )
25		simplr 754	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> U e. J )
26		eldifi 3585	. . . . . . . 8 |- ( w e. ( X \ U ) -> w e. X )
27	26	adantl 466	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> w e. X )
28	3	kqfvima 19434	. . . . . . 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 1219	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> ( w e. U <-> ( F ` w ) e. ( F " U ) ) )
30	23, 29	mtbid 300	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. J ) /\ w e. ( X \ U ) ) -> -. ( F ` w ) e. ( F " U ) )
31	30	ralrimiva 2829	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> A. w e. ( X \ U ) -. ( F ` w ) e. ( F " U ) )
32		difss 3590	. . . . 5 |- ( X \ U ) C_ X
33		eleq1 2526	. . . . . . 7 |- ( z = ( F ` w ) -> ( z e. ( F " U ) <-> ( F ` w ) e. ( F " U ) ) )
34	33	notbid 294	. . . . . 6 |- ( z = ( F ` w ) -> ( -. z e. ( F " U ) <-> -. ( F ` w ) e. ( F " U ) ) )
35	34	ralima 6065	. . . . 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 662	. . . 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 232	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> A. z e. ( F " ( X \ U ) ) -. z e. ( F " U ) )
38		disjr 3827	. . 3 |- ( ( ( F " U ) i^i ( F " ( X \ U ) ) ) = (/) <-> A. z e. ( F " ( X \ U ) ) -. z e. ( F " U ) )
39	37, 38	sylibr 212	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( F " U ) i^i ( F " ( X \ U ) ) ) = (/) )
40		sseq0 3776	. 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 661	1 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( F " U ) i^i ( F " ( A \ U ) ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   A.wral 2798   {crab 2802   \ cdif 3432   i^i cin 3434   C_ wss 3435   (/)c0 3744   |-> cmpt 4457   dom cdm 4947   |` cres 4949   "cima 4950   Fn wfn 5520   ` cfv 5525  TopOnctopon 18630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-topon 18637

This theorem is referenced by:  kqcldsat  19437  regr1lem  19443