Theorem restdis 14843

Description: A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
restdis	|- ( ( A e. V /\ B C_ A ) -> ( ~P A ↾t B ) = ~P B )

Proof of Theorem restdis

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		distop 14744	. . . . 5 |- ( A e. V -> ~P A e. Top )
2	1	adantr 276	. . . 4 |- ( ( A e. V /\ B C_ A ) -> ~P A e. Top )
3		elpw2g 4239	. . . . 5 |- ( A e. V -> ( B e. ~P A <-> B C_ A ) )
4	3	biimpar 297	. . . 4 |- ( ( A e. V /\ B C_ A ) -> B e. ~P A )
5		restopn2 14842	. . . 4 |- ( ( ~P A e. Top /\ B e. ~P A ) -> ( x e. ( ~P A ↾t B ) <-> ( x e. ~P A /\ x C_ B ) ) )
6	2, 4, 5	syl2anc 411	. . 3 |- ( ( A e. V /\ B C_ A ) -> ( x e. ( ~P A ↾t B ) <-> ( x e. ~P A /\ x C_ B ) ) )
7		velpw 3656	. . . 4 |- ( x e. ~P B <-> x C_ B )
8		sstr 3232	. . . . . . . 8 |- ( ( x C_ B /\ B C_ A ) -> x C_ A )
9	8	expcom 116	. . . . . . 7 |- ( B C_ A -> ( x C_ B -> x C_ A ) )
10	9	adantl 277	. . . . . 6 |- ( ( A e. V /\ B C_ A ) -> ( x C_ B -> x C_ A ) )
11		velpw 3656	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
12	10, 11	imbitrrdi 162	. . . . 5 |- ( ( A e. V /\ B C_ A ) -> ( x C_ B -> x e. ~P A ) )
13	12	pm4.71rd 394	. . . 4 |- ( ( A e. V /\ B C_ A ) -> ( x C_ B <-> ( x e. ~P A /\ x C_ B ) ) )
14	7, 13	bitrid 192	. . 3 |- ( ( A e. V /\ B C_ A ) -> ( x e. ~P B <-> ( x e. ~P A /\ x C_ B ) ) )
15	6, 14	bitr4d 191	. 2 |- ( ( A e. V /\ B C_ A ) -> ( x e. ( ~P A ↾t B ) <-> x e. ~P B ) )
16	15	eqrdv 2227	1 |- ( ( A e. V /\ B C_ A ) -> ( ~P A ↾t B ) = ~P B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   C_ wss 3197   ~Pcpw 3649  (class class class)co 5994   ↾_t crest 13258   Topctop 14656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-rest 13260  df-topgen 13279  df-top 14657  df-topon 14670  df-bases 14702

This theorem is referenced by: (None)