Theorem restabs 12126

Description: Equivalence of being a subspace of a subspace and being a subspace of the original. (Contributed by Jeff Hankins, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)

Assertion

Ref	Expression
restabs	|- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( ( J ↾t T ) ↾t S ) = ( J ↾t S ) )

Proof of Theorem restabs

Step	Hyp	Ref	Expression
1		simp1 949	. . 3 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> J e. V )
2		simp3 951	. . 3 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> T e. W )
3		ssexg 4007	. . . 4 |- ( ( S C_ T /\ T e. W ) -> S e. _V )
4	3	3adant1 967	. . 3 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> S e. _V )
5		restco 12125	. . 3 |- ( ( J e. V /\ T e. W /\ S e. _V ) -> ( ( J ↾t T ) ↾t S ) = ( J ↾t ( T i^i S ) ) )
6	1, 2, 4, 5	syl3anc 1184	. 2 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( ( J ↾t T ) ↾t S ) = ( J ↾t ( T i^i S ) ) )
7		simp2 950	. . . 4 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> S C_ T )
8		sseqin2 3242	. . . 4 |- ( S C_ T <-> ( T i^i S ) = S )
9	7, 8	sylib 121	. . 3 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( T i^i S ) = S )
10	9	oveq2d 5722	. 2 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( J ↾t ( T i^i S ) ) = ( J ↾t S ) )
11	6, 10	eqtrd 2132	1 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( ( J ↾t T ) ↾t S ) = ( J ↾t S ) )

Syntax hints:   -> wi 4   /\ w3a 930   = wceq 1299   e. wcel 1448   _Vcvv 2641   i^i cin 3020   C_ wss 3021  (class class class)co 5706   ↾_t crest 11902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-rest 11904

This theorem is referenced by:  rerestcntop  12469