Theorem bj-restpw 33045

Description: The elementwise intersection on a powerset is the powerset of the intersection. This allows to prove for instance that the topology induced on a subset by the discrete topology is the discrete topology on that subset. See also restdis 20982 (which uses distop 20799 and restopn2 20981). (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-restpw	|- ( ( Y e. V /\ A e. W ) -> ( ~P Y ↾t A ) = ~P ( Y i^i A ) )

Proof of Theorem bj-restpw

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 4850	. . . 4 |- ( Y e. V -> ~P Y e. _V )
2		elrest 16088	. . . 4 |- ( ( ~P Y e. _V /\ A e. W ) -> ( x e. ( ~P Y ↾t A ) <-> E. y e. ~P Y x = ( y i^i A ) ) )
3	1, 2	sylan 488	. . 3 |- ( ( Y e. V /\ A e. W ) -> ( x e. ( ~P Y ↾t A ) <-> E. y e. ~P Y x = ( y i^i A ) ) )
4		selpw 4165	. . . . . . 7 |- ( y e. ~P Y <-> y C_ Y )
5	4	anbi1i 731	. . . . . 6 |- ( ( y e. ~P Y /\ x = ( y i^i A ) ) <-> ( y C_ Y /\ x = ( y i^i A ) ) )
6	5	exbii 1774	. . . . 5 |- ( E. y ( y e. ~P Y /\ x = ( y i^i A ) ) <-> E. y ( y C_ Y /\ x = ( y i^i A ) ) )
7		sstr2 3610	. . . . . . . . . 10 |- ( x C_ y -> ( y C_ Y -> x C_ Y ) )
8	7	com12 32	. . . . . . . . 9 |- ( y C_ Y -> ( x C_ y -> x C_ Y ) )
9		inss1 3833	. . . . . . . . . 10 |- ( y i^i A ) C_ y
10		sseq1 3626	. . . . . . . . . 10 |- ( x = ( y i^i A ) -> ( x C_ y <-> ( y i^i A ) C_ y ) )
11	9, 10	mpbiri 248	. . . . . . . . 9 |- ( x = ( y i^i A ) -> x C_ y )
12	8, 11	impel 485	. . . . . . . 8 |- ( ( y C_ Y /\ x = ( y i^i A ) ) -> x C_ Y )
13		inss2 3834	. . . . . . . . . 10 |- ( y i^i A ) C_ A
14		sseq1 3626	. . . . . . . . . 10 |- ( x = ( y i^i A ) -> ( x C_ A <-> ( y i^i A ) C_ A ) )
15	13, 14	mpbiri 248	. . . . . . . . 9 |- ( x = ( y i^i A ) -> x C_ A )
16	15	adantl 482	. . . . . . . 8 |- ( ( y C_ Y /\ x = ( y i^i A ) ) -> x C_ A )
17	12, 16	ssind 3837	. . . . . . 7 |- ( ( y C_ Y /\ x = ( y i^i A ) ) -> x C_ ( Y i^i A ) )
18	17	exlimiv 1858	. . . . . 6 |- ( E. y ( y C_ Y /\ x = ( y i^i A ) ) -> x C_ ( Y i^i A ) )
19		inss1 3833	. . . . . . . 8 |- ( Y i^i A ) C_ Y
20		sstr2 3610	. . . . . . . 8 |- ( x C_ ( Y i^i A ) -> ( ( Y i^i A ) C_ Y -> x C_ Y ) )
21	19, 20	mpi 20	. . . . . . 7 |- ( x C_ ( Y i^i A ) -> x C_ Y )
22		inss2 3834	. . . . . . . 8 |- ( Y i^i A ) C_ A
23		sstr2 3610	. . . . . . . 8 |- ( x C_ ( Y i^i A ) -> ( ( Y i^i A ) C_ A -> x C_ A ) )
24	22, 23	mpi 20	. . . . . . 7 |- ( x C_ ( Y i^i A ) -> x C_ A )
25		ssid 3624	. . . . . . . . . . 11 |- x C_ x
26	25	a1i 11	. . . . . . . . . 10 |- ( x C_ A -> x C_ x )
27		id 22	. . . . . . . . . 10 |- ( x C_ A -> x C_ A )
28	26, 27	ssind 3837	. . . . . . . . 9 |- ( x C_ A -> x C_ ( x i^i A ) )
29		inss1 3833	. . . . . . . . . 10 |- ( x i^i A ) C_ x
30	29	a1i 11	. . . . . . . . 9 |- ( x C_ A -> ( x i^i A ) C_ x )
31	28, 30	eqssd 3620	. . . . . . . 8 |- ( x C_ A -> x = ( x i^i A ) )
32		vex 3203	. . . . . . . . 9 |- x e. _V
33		sseq1 3626	. . . . . . . . . 10 |- ( y = x -> ( y C_ Y <-> x C_ Y ) )
34		ineq1 3807	. . . . . . . . . . 11 |- ( y = x -> ( y i^i A ) = ( x i^i A ) )
35	34	eqeq2d 2632	. . . . . . . . . 10 |- ( y = x -> ( x = ( y i^i A ) <-> x = ( x i^i A ) ) )
36	33, 35	anbi12d 747	. . . . . . . . 9 |- ( y = x -> ( ( y C_ Y /\ x = ( y i^i A ) ) <-> ( x C_ Y /\ x = ( x i^i A ) ) ) )
37	32, 36	spcev 3300	. . . . . . . 8 |- ( ( x C_ Y /\ x = ( x i^i A ) ) -> E. y ( y C_ Y /\ x = ( y i^i A ) ) )
38	31, 37	sylan2 491	. . . . . . 7 |- ( ( x C_ Y /\ x C_ A ) -> E. y ( y C_ Y /\ x = ( y i^i A ) ) )
39	21, 24, 38	syl2anc 693	. . . . . 6 |- ( x C_ ( Y i^i A ) -> E. y ( y C_ Y /\ x = ( y i^i A ) ) )
40	18, 39	impbii 199	. . . . 5 |- ( E. y ( y C_ Y /\ x = ( y i^i A ) ) <-> x C_ ( Y i^i A ) )
41	6, 40	bitri 264	. . . 4 |- ( E. y ( y e. ~P Y /\ x = ( y i^i A ) ) <-> x C_ ( Y i^i A ) )
42		df-rex 2918	. . . 4 |- ( E. y e. ~P Y x = ( y i^i A ) <-> E. y ( y e. ~P Y /\ x = ( y i^i A ) ) )
43		selpw 4165	. . . 4 |- ( x e. ~P ( Y i^i A ) <-> x C_ ( Y i^i A ) )
44	41, 42, 43	3bitr4i 292	. . 3 |- ( E. y e. ~P Y x = ( y i^i A ) <-> x e. ~P ( Y i^i A ) )
45	3, 44	syl6bb 276	. 2 |- ( ( Y e. V /\ A e. W ) -> ( x e. ( ~P Y ↾t A ) <-> x e. ~P ( Y i^i A ) ) )
46	45	eqrdv 2620	1 |- ( ( Y e. V /\ A e. W ) -> ( ~P Y ↾t A ) = ~P ( Y i^i A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158  (class class class)co 6650   ↾_t crest 16081

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rest 16083

This theorem is referenced by: (None)