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Theorem restin 22892
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 7734 . . . . 5 (𝐽𝑉 𝐽 ∈ V)
31, 2eqeltrid 2835 . . . 4 (𝐽𝑉𝑋 ∈ V)
43adantr 479 . . 3 ((𝐽𝑉𝐴𝑊) → 𝑋 ∈ V)
5 restco 22890 . . . 4 ((𝐽𝑉𝑋 ∈ V ∧ 𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
653com23 1124 . . 3 ((𝐽𝑉𝐴𝑊𝑋 ∈ V) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
74, 6mpd3an3 1460 . 2 ((𝐽𝑉𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
81restid 17385 . . . 4 (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)
98adantr 479 . . 3 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝑋) = 𝐽)
109oveq1d 7428 . 2 ((𝐽𝑉𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t 𝐴))
11 incom 4202 . . . 4 (𝑋𝐴) = (𝐴𝑋)
1211oveq2i 7424 . . 3 (𝐽t (𝑋𝐴)) = (𝐽t (𝐴𝑋))
1312a1i 11 . 2 ((𝐽𝑉𝐴𝑊) → (𝐽t (𝑋𝐴)) = (𝐽t (𝐴𝑋)))
147, 10, 133eqtr3d 2778 1 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = (𝐽t (𝐴𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  Vcvv 3472  cin 3948   cuni 4909  (class class class)co 7413  t crest 17372
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7416  df-oprab 7417  df-mpo 7418  df-rest 17374
This theorem is referenced by:  restuni2  22893  cnrest2r  23013  cnrmi  23086  restcnrm  23088  resthauslem  23089  imacmp  23123  fiuncmp  23130  kgeni  23263  ressxms  24256  ptrest  36792  restuni6  44114
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