Theorem restdis 20982

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 20799	. . . . 5 |- ( A e. V -> ~P A e. Top )
2	1	adantr 481	. . . 4 |- ( ( A e. V /\ B C_ A ) -> ~P A e. Top )
3		elpw2g 4827	. . . . 5 |- ( A e. V -> ( B e. ~P A <-> B C_ A ) )
4	3	biimpar 502	. . . 4 |- ( ( A e. V /\ B C_ A ) -> B e. ~P A )
5		restopn2 20981	. . . 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 693	. . 3 |- ( ( A e. V /\ B C_ A ) -> ( x e. ( ~P A ↾t B ) <-> ( x e. ~P A /\ x C_ B ) ) )
7		selpw 4165	. . . 4 |- ( x e. ~P B <-> x C_ B )
8		sstr 3611	. . . . . . . 8 |- ( ( x C_ B /\ B C_ A ) -> x C_ A )
9	8	expcom 451	. . . . . . 7 |- ( B C_ A -> ( x C_ B -> x C_ A ) )
10	9	adantl 482	. . . . . 6 |- ( ( A e. V /\ B C_ A ) -> ( x C_ B -> x C_ A ) )
11		selpw 4165	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
12	10, 11	syl6ibr 242	. . . . 5 |- ( ( A e. V /\ B C_ A ) -> ( x C_ B -> x e. ~P A ) )
13	12	pm4.71rd 667	. . . 4 |- ( ( A e. V /\ B C_ A ) -> ( x C_ B <-> ( x e. ~P A /\ x C_ B ) ) )
14	7, 13	syl5bb 272	. . 3 |- ( ( A e. V /\ B C_ A ) -> ( x e. ~P B <-> ( x e. ~P A /\ x C_ B ) ) )
15	6, 14	bitr4d 271	. 2 |- ( ( A e. V /\ B C_ A ) -> ( x e. ( ~P A ↾t B ) <-> x e. ~P B ) )
16	15	eqrdv 2620	1 |- ( ( A e. V /\ B C_ A ) -> ( ~P A ↾t B ) = ~P B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158  (class class class)co 6650   ↾_t crest 16081   Topctop 20698

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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750

This theorem is referenced by:  dislly  21300  xkopt  21458