Theorem restdis 15285

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 15186	. . . . 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 4292	. . . . 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 15284	. . . 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 415	. . 3 |- ( ( A e. V /\ B C_ A ) -> ( x e. ( ~P A ↾t B ) <-> ( x e. ~P A /\ x C_ B ) ) )
7		velpw 3695	. . . 4 |- ( x e. ~P B <-> x C_ B )
8		sstr 3256	. . . . . . . 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 3695	. . . . . 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 398	. . . 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 2236	1 |- ( ( A e. V /\ B C_ A ) -> ( ~P A ↾t B ) = ~P B )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   C_ wss 3220   ~Pcpw 3688  (class class class)co 6085   ↾_t crest 13593   Topctop 15098

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-rest 13595  df-topgen 13614  df-top 15099  df-topon 15112  df-bases 15144

This theorem is used by: (None)