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Theorem restid 11913
Description: The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
Hypothesis
Ref Expression
restid.1 𝑋 = 𝐽
Assertion
Ref Expression
restid (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)

Proof of Theorem restid
StepHypRef Expression
1 restid.1 . . 3 𝑋 = 𝐽
2 uniexg 4299 . . 3 (𝐽𝑉 𝐽 ∈ V)
31, 2syl5eqel 2186 . 2 (𝐽𝑉𝑋 ∈ V)
41eqimss2i 3104 . . 3 𝐽𝑋
5 sspwuni 3843 . . 3 (𝐽 ⊆ 𝒫 𝑋 𝐽𝑋)
64, 5mpbir 145 . 2 𝐽 ⊆ 𝒫 𝑋
7 restid2 11911 . 2 ((𝑋 ∈ V ∧ 𝐽 ⊆ 𝒫 𝑋) → (𝐽t 𝑋) = 𝐽)
83, 6, 7sylancl 407 1 (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  wcel 1448  Vcvv 2641  wss 3021  𝒫 cpw 3457   cuni 3683  (class class class)co 5706  t crest 11902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-rest 11904
This theorem is referenced by:  toponrestid  11970  restin  12127
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