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Theorem restabs 20779
 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 1054 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝐽𝑉)
2 simp3 1056 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑇𝑊)
3 ssexg 4732 . . . 4 ((𝑆𝑇𝑇𝑊) → 𝑆 ∈ V)
433adant1 1072 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑆 ∈ V)
5 restco 20778 . . 3 ((𝐽𝑉𝑇𝑊𝑆 ∈ V) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t (𝑇𝑆)))
61, 2, 4, 5syl3anc 1318 . 2 ((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t (𝑇𝑆)))
7 simp2 1055 . . . 4 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑆𝑇)
8 sseqin2 3779 . . . 4 (𝑆𝑇 ↔ (𝑇𝑆) = 𝑆)
97, 8sylib 207 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → (𝑇𝑆) = 𝑆)
109oveq2d 6565 . 2 ((𝐽𝑉𝑆𝑇𝑇𝑊) → (𝐽t (𝑇𝑆)) = (𝐽t 𝑆))
116, 10eqtrd 2644 1 ((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  (class class class)co 6549   ↾t crest 15904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rest 15906 This theorem is referenced by:  restcnrm  20976  fiuncmp  21017  subislly  21094  restnlly  21095  islly2  21097  llyrest  21098  nllyrest  21099  llyidm  21101  nllyidm  21102  cldllycmp  21108  txkgen  21265  rerest  22415  xrrest  22418  cnmpt2pc  22535  cnheiborlem  22561  pcoass  22632  limcres  23456  perfdvf  23473  dvreslem  23479  dvres2lem  23480  dvaddbr  23507  dvmulbr  23508  dvcnvrelem2  23585  psercn  23984  abelth  23999  cxpcn2  24287  cxpcn3  24289  lmlimxrge0  29322  pnfneige0  29325  cvmsss2  30510  cvmliftlem8  30528  cvmliftlem10  30530  cvmlift2lem9  30547  ivthALT  31500  limcresiooub  38709  limcresioolb  38710  cncfuni  38772  cncfiooicclem1  38779  itgsubsticclem  38867  dirkercncflem4  38999  fourierdlem32  39032  fourierdlem33  39033  fourierdlem62  39061  fouriersw  39124  smfco  39687
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