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Theorem restin 12931
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 4422 . . . . 5 (𝐽𝑉 𝐽 ∈ V)
31, 2eqeltrid 2257 . . . 4 (𝐽𝑉𝑋 ∈ V)
43adantr 274 . . 3 ((𝐽𝑉𝐴𝑊) → 𝑋 ∈ V)
5 restco 12929 . . . 4 ((𝐽𝑉𝑋 ∈ V ∧ 𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
653com23 1204 . . 3 ((𝐽𝑉𝐴𝑊𝑋 ∈ V) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
74, 6mpd3an3 1333 . 2 ((𝐽𝑉𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t (𝑋𝐴)))
81restid 12579 . . . 4 (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)
98adantr 274 . . 3 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝑋) = 𝐽)
109oveq1d 5866 . 2 ((𝐽𝑉𝐴𝑊) → ((𝐽t 𝑋) ↾t 𝐴) = (𝐽t 𝐴))
11 incom 3319 . . . 4 (𝑋𝐴) = (𝐴𝑋)
1211oveq2i 5862 . . 3 (𝐽t (𝑋𝐴)) = (𝐽t (𝐴𝑋))
1312a1i 9 . 2 ((𝐽𝑉𝐴𝑊) → (𝐽t (𝑋𝐴)) = (𝐽t (𝐴𝑋)))
147, 10, 133eqtr3d 2211 1 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = (𝐽t (𝐴𝑋)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  Vcvv 2730  cin 3120   cuni 3794  (class class class)co 5851  t crest 12568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-rest 12570
This theorem is referenced by:  restuni2  12932  cnrest2r  12992
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