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

Theorem restin 21767
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 7456 . . . . 5 (𝐽𝑉 𝐽 ∈ V)
31, 2eqeltrid 2920 . . . 4 (𝐽𝑉𝑋 ∈ V)
43adantr 484 . . 3 ((𝐽𝑉𝐴𝑊) → 𝑋 ∈ V)
5 restco 21765 . . . 4 ((𝐽𝑉𝑋 ∈ V ∧ 𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
653com23 1123 . . 3 ((𝐽𝑉𝐴𝑊𝑋 ∈ V) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
74, 6mpd3an3 1459 . 2 ((𝐽𝑉𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
81restid 16703 . . . 4 (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)
98adantr 484 . . 3 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝑋) = 𝐽)
109oveq1d 7160 . 2 ((𝐽𝑉𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t 𝐴))
11 incom 4162 . . . 4 (𝑋𝐴) = (𝐴𝑋)
1211oveq2i 7156 . . 3 (𝐽t (𝑋𝐴)) = (𝐽t (𝐴𝑋))
1312a1i 11 . 2 ((𝐽𝑉𝐴𝑊) → (𝐽t (𝑋𝐴)) = (𝐽t (𝐴𝑋)))
147, 10, 133eqtr3d 2867 1 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = (𝐽t (𝐴𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  cin 3918   cuni 4824  (class class class)co 7145  t crest 16690
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-rest 16692
This theorem is referenced by:  restuni2  21768  cnrest2r  21888  cnrmi  21961  restcnrm  21963  resthauslem  21964  imacmp  21998  fiuncmp  22005  kgeni  22138  ressxms  23128  ptrest  34966  restuni6  41616
  Copyright terms: Public domain W3C validator