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Theorem restabs 21467
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 1116 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝐽𝑉)
2 simp3 1118 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑇𝑊)
3 ssexg 5077 . . . 4 ((𝑆𝑇𝑇𝑊) → 𝑆 ∈ V)
433adant1 1110 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑆 ∈ V)
5 restco 21466 . . 3 ((𝐽𝑉𝑇𝑊𝑆 ∈ V) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t (𝑇𝑆)))
61, 2, 4, 5syl3anc 1351 . 2 ((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t (𝑇𝑆)))
7 simp2 1117 . . . 4 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑆𝑇)
8 sseqin2 4074 . . . 4 (𝑆𝑇 ↔ (𝑇𝑆) = 𝑆)
97, 8sylib 210 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → (𝑇𝑆) = 𝑆)
109oveq2d 6986 . 2 ((𝐽𝑉𝑆𝑇𝑇𝑊) → (𝐽t (𝑇𝑆)) = (𝐽t 𝑆))
116, 10eqtrd 2808 1 ((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1068   = wceq 1507  wcel 2048  Vcvv 3409  cin 3824  wss 3825  (class class class)co 6970  t crest 16540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-rest 16542
This theorem is referenced by:  restcnrm  21664  fiuncmp  21706  subislly  21783  restnlly  21784  islly2  21786  llyrest  21787  nllyrest  21788  llyidm  21790  nllyidm  21791  cldllycmp  21797  txkgen  21954  rerest  23105  xrrest  23108  cnmpopc  23225  cnheiborlem  23251  pcoass  23321  limcres  24177  perfdvf  24194  dvreslem  24200  dvres2lem  24201  dvaddbr  24228  dvmulbr  24229  dvcnvrelem2  24308  psercn  24707  abelth  24722  cxpcn2  25018  cxpcn3  25020  lmlimxrge0  30792  pnfneige0  30795  cvmsss2  32066  cvmliftlem8  32084  cvmliftlem10  32086  cvmlift2lem9  32103  ivthALT  33144  limcresiooub  41300  limcresioolb  41301  cncfuni  41545  cncfiooicclem1  41552  itgsubsticclem  41636  dirkercncflem4  41768  fourierdlem32  41801  fourierdlem33  41802  fourierdlem62  41830  fouriersw  41893  smfco  42454
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