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Theorem restin 23195
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 7775 . . . . 5 (𝐽𝑉 𝐽 ∈ V)
31, 2eqeltrid 2848 . . . 4 (𝐽𝑉𝑋 ∈ V)
43adantr 480 . . 3 ((𝐽𝑉𝐴𝑊) → 𝑋 ∈ V)
5 restco 23193 . . . 4 ((𝐽𝑉𝑋 ∈ V ∧ 𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
653com23 1126 . . 3 ((𝐽𝑉𝐴𝑊𝑋 ∈ V) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
74, 6mpd3an3 1462 . 2 ((𝐽𝑉𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
81restid 17493 . . . 4 (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)
98adantr 480 . . 3 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝑋) = 𝐽)
109oveq1d 7463 . 2 ((𝐽𝑉𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t 𝐴))
11 incom 4230 . . . 4 (𝑋𝐴) = (𝐴𝑋)
1211oveq2i 7459 . . 3 (𝐽t (𝑋𝐴)) = (𝐽t (𝐴𝑋))
1312a1i 11 . 2 ((𝐽𝑉𝐴𝑊) → (𝐽t (𝑋𝐴)) = (𝐽t (𝐴𝑋)))
147, 10, 133eqtr3d 2788 1 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = (𝐽t (𝐴𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  Vcvv 3488  cin 3975   cuni 4931  (class class class)co 7448  t crest 17480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-rest 17482
This theorem is referenced by:  restuni2  23196  cnrest2r  23316  cnrmi  23389  restcnrm  23391  resthauslem  23392  imacmp  23426  fiuncmp  23433  kgeni  23566  ressxms  24559  ptrest  37579  restuni6  45024
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