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Theorem restabs 14152
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 )  ->  (
( Jt  T )t  S )  =  ( Jt  S ) )

Proof of Theorem restabs
StepHypRef Expression
1 simp1 999 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  J  e.  V )
2 simp3 1001 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  T  e.  W )
3 ssexg 4157 . . . 4  |-  ( ( S  C_  T  /\  T  e.  W )  ->  S  e.  _V )
433adant1 1017 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  S  e.  _V )
5 restco 14151 . . 3  |-  ( ( J  e.  V  /\  T  e.  W  /\  S  e.  _V )  ->  ( ( Jt  T )t  S )  =  ( Jt  ( T  i^i  S ) ) )
61, 2, 4, 5syl3anc 1249 . 2  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  (
( Jt  T )t  S )  =  ( Jt  ( T  i^i  S
) ) )
7 simp2 1000 . . . 4  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  S  C_  T )
8 sseqin2 3369 . . . 4  |-  ( S 
C_  T  <->  ( T  i^i  S )  =  S )
97, 8sylib 122 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  ( T  i^i  S )  =  S )
109oveq2d 5913 . 2  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  ( Jt  ( T  i^i  S ) )  =  ( Jt  S ) )
116, 10eqtrd 2222 1  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  (
( Jt  T )t  S )  =  ( Jt  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 980    = wceq 1364    e. wcel 2160   _Vcvv 2752    i^i cin 3143    C_ wss 3144  (class class class)co 5897   ↾t crest 12747
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-rest 12749
This theorem is referenced by:  rerestcntop  14527
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