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Theorem restabs 21780
 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 ((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t 𝑆))

Proof of Theorem restabs
StepHypRef Expression
1 simp1 1133 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝐽𝑉)
2 simp3 1135 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑇𝑊)
3 ssexg 5192 . . . 4 ((𝑆𝑇𝑇𝑊) → 𝑆 ∈ V)
433adant1 1127 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑆 ∈ V)
5 restco 21779 . . 3 ((𝐽𝑉𝑇𝑊𝑆 ∈ V) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t (𝑇𝑆)))
61, 2, 4, 5syl3anc 1368 . 2 ((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t (𝑇𝑆)))
7 simp2 1134 . . . 4 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑆𝑇)
8 sseqin2 4142 . . . 4 (𝑆𝑇 ↔ (𝑇𝑆) = 𝑆)
97, 8sylib 221 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → (𝑇𝑆) = 𝑆)
109oveq2d 7152 . 2 ((𝐽𝑉𝑆𝑇𝑇𝑊) → (𝐽t (𝑇𝑆)) = (𝐽t 𝑆))
116, 10eqtrd 2833 1 ((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  (class class class)co 7136   ↾t crest 16689 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-rest 16691 This theorem is referenced by:  restcnrm  21977  fiuncmp  22019  subislly  22096  restnlly  22097  islly2  22099  llyrest  22100  nllyrest  22101  llyidm  22103  nllyidm  22104  cldllycmp  22110  txkgen  22267  rerest  23419  xrrest  23422  cnmpopc  23543  cnheiborlem  23569  pcoass  23639  limcres  24499  perfdvf  24516  dvreslem  24522  dvres2lem  24523  dvaddbr  24551  dvmulbr  24552  dvcnvrelem2  24631  psercn  25031  abelth  25046  cxpcn2  25345  cxpcn3  25347  lmlimxrge0  31316  pnfneige0  31319  cvmsss2  32649  cvmliftlem8  32667  cvmliftlem10  32669  cvmlift2lem9  32686  ivthALT  33811  limcresiooub  42327  limcresioolb  42328  cncfuni  42571  cncfiooicclem1  42578  itgsubsticclem  42660  dirkercncflem4  42791  fourierdlem32  42824  fourierdlem33  42825  fourierdlem62  42853  fouriersw  42916  smfco  43477
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