Theorem restdis 14978

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 14879	. . . . 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 4251	. . . . 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 14977	. . . 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 3663	. . . 4 |- ( x e. ~P B <-> x C_ B )
8		sstr 3236	. . . . . . . 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 3663	. . . . . 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 2229	1 |- ( ( A e. V /\ B C_ A ) -> ( ~P A ↾t B ) = ~P B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   C_ wss 3201   ~Pcpw 3656  (class class class)co 6028   ↾_t crest 13385   Topctop 14791

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-rest 13387  df-topgen 13406  df-top 14792  df-topon 14805  df-bases 14837

This theorem is referenced by: (None)