MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  restin Structured version   Visualization version   GIF version

Theorem restin 23174
Description: When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.)
Hypothesis
Ref Expression
restin.1 𝑋 = 𝐽
Assertion
Ref Expression
restin ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = (𝐽t (𝐴𝑋)))

Proof of Theorem restin
StepHypRef Expression
1 restin.1 . . . . 5 𝑋 = 𝐽
2 uniexg 7760 . . . . 5 (𝐽𝑉 𝐽 ∈ V)
31, 2eqeltrid 2845 . . . 4 (𝐽𝑉𝑋 ∈ V)
43adantr 480 . . 3 ((𝐽𝑉𝐴𝑊) → 𝑋 ∈ V)
5 restco 23172 . . . 4 ((𝐽𝑉𝑋 ∈ V ∧ 𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
653com23 1127 . . 3 ((𝐽𝑉𝐴𝑊𝑋 ∈ V) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
74, 6mpd3an3 1464 . 2 ((𝐽𝑉𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
81restid 17478 . . . 4 (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)
98adantr 480 . . 3 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝑋) = 𝐽)
109oveq1d 7446 . 2 ((𝐽𝑉𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t 𝐴))
11 incom 4209 . . . 4 (𝑋𝐴) = (𝐴𝑋)
1211oveq2i 7442 . . 3 (𝐽t (𝑋𝐴)) = (𝐽t (𝐴𝑋))
1312a1i 11 . 2 ((𝐽𝑉𝐴𝑊) → (𝐽t (𝑋𝐴)) = (𝐽t (𝐴𝑋)))
147, 10, 133eqtr3d 2785 1 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = (𝐽t (𝐴𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950   cuni 4907  (class class class)co 7431  t crest 17465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17467
This theorem is referenced by:  restuni2  23175  cnrest2r  23295  cnrmi  23368  restcnrm  23370  resthauslem  23371  imacmp  23405  fiuncmp  23412  kgeni  23545  ressxms  24538  ptrest  37626  restuni6  45127
  Copyright terms: Public domain W3C validator