Theorem restabs 12825

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 987	. . 3 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> J e. V )
2		simp3 989	. . 3 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> T e. W )
3		ssexg 4121	. . . 4 |- ( ( S C_ T /\ T e. W ) -> S e. _V )
4	3	3adant1 1005	. . 3 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> S e. _V )
5		restco 12824	. . 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 1228	. 2 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( ( J ↾t T ) ↾t S ) = ( J ↾t ( T i^i S ) ) )
7		simp2 988	. . . 4 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> S C_ T )
8		sseqin2 3341	. . . 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 5858	. 2 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( J ↾t ( T i^i S ) ) = ( J ↾t S ) )
11	6, 10	eqtrd 2198	1 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( ( J ↾t T ) ↾t S ) = ( J ↾t S ) )

Syntax hints:   -> wi 4   /\ w3a 968   = wceq 1343   e. wcel 2136   _Vcvv 2726   i^i cin 3115   C_ wss 3116  (class class class)co 5842   ↾_t crest 12556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-rest 12558

This theorem is referenced by:  rerestcntop  13200