Theorem kqdisj 19965

Description: A version of imain 5662 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 5497	. . . . 5 |- ( F " dom ( F |` ( A \ U ) ) ) = ( F " ( A \ U ) )
2		dmres 5292	. . . . . . 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 19958	. . . . . . . . . 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 5678	. . . . . . . . 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 3700	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( ( A \ U ) i^i dom F ) = ( ( A \ U ) i^i X ) )
9	2, 8	syl5eq 2520	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> dom ( F |` ( A \ U ) ) = ( ( A \ U ) i^i X ) )
10	9	imaeq2d 5335	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " dom ( F |` ( A \ U ) ) ) = ( F " ( ( A \ U ) i^i X ) ) )
11	1, 10	syl5eqr 2522	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " ( A \ U ) ) = ( F " ( ( A \ U ) i^i X ) ) )
12		indif1 3742	. . . . . 6 |- ( ( A \ U ) i^i X ) = ( ( A i^i X ) \ U )
13		inss2 3719	. . . . . . 7 |- ( A i^i X ) C_ X
14		ssdif 3639	. . . . . . 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 3534	. . . . 5 |- ( ( A \ U ) i^i X ) C_ ( X \ U )
17		imass2 5370	. . . . 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 3538	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> ( F " ( A \ U ) ) C_ ( F " ( X \ U ) ) )
20		sslin 3724	. . 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 3627	. . . . . . 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 3626	. . . . . . . 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 19963	. . . . . . 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 1228	. . . . . 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 2878	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> A. w e. ( X \ U ) -. ( F ` w ) e. ( F " U ) )
32		difss 3631	. . . . 5 |- ( X \ U ) C_ X
33		eleq1 2539	. . . . . . 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 6138	. . . . 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 3868	. . 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 3817	. 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 1379   e. wcel 1767   A.wral 2814   {crab 2818   \ cdif 3473   i^i cin 3475   C_ wss 3476   (/)c0 3785   |-> cmpt 4505   dom cdm 4999   |` cres 5001   "cima 5002   Fn wfn 5581   ` cfv 5586  TopOnctopon 19159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-topon 19166

This theorem is referenced by:  kqcldsat  19966  regr1lem  19972