Theorem restabs 14966

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 1024	. . 3 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> J e. V )
2		simp3 1026	. . 3 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> T e. W )
3		ssexg 4233	. . . 4 |- ( ( S C_ T /\ T e. W ) -> S e. _V )
4	3	3adant1 1042	. . 3 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> S e. _V )
5		restco 14965	. . 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 1274	. 2 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( ( J ↾t T ) ↾t S ) = ( J ↾t ( T i^i S ) ) )
7		simp2 1025	. . . 4 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> S C_ T )
8		sseqin2 3428	. . . 4 |- ( S C_ T <-> ( T i^i S ) = S )
9	7, 8	sylib 122	. . 3 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( T i^i S ) = S )
10	9	oveq2d 6044	. 2 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( J ↾t ( T i^i S ) ) = ( J ↾t S ) )
11	6, 10	eqtrd 2264	1 |- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( ( J ↾t T ) ↾t S ) = ( J ↾t S ) )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2202   _Vcvv 2803   i^i cin 3200   C_ wss 3201  (class class class)co 6028   ↾_t crest 13383

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 13385

This theorem is referenced by:  rerestcntop  15349  rerest  15351