Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  restid2 Structured version   Visualization version   GIF version

Theorem restid2 16775
 Description: The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
restid2 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝐽t 𝐴) = 𝐽)

Proof of Theorem restid2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pwexg 5251 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
21adantr 484 . . . 4 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → 𝒫 𝐴 ∈ V)
3 simpr 488 . . . 4 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → 𝐽 ⊆ 𝒫 𝐴)
42, 3ssexd 5198 . . 3 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → 𝐽 ∈ V)
5 simpl 486 . . 3 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → 𝐴𝑉)
6 restval 16771 . . 3 ((𝐽 ∈ V ∧ 𝐴𝑉) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
74, 5, 6syl2anc 587 . 2 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
83sselda 3894 . . . . . . . 8 (((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) ∧ 𝑥𝐽) → 𝑥 ∈ 𝒫 𝐴)
98elpwid 4508 . . . . . . 7 (((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) ∧ 𝑥𝐽) → 𝑥𝐴)
10 df-ss 3877 . . . . . . 7 (𝑥𝐴 ↔ (𝑥𝐴) = 𝑥)
119, 10sylib 221 . . . . . 6 (((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) ∧ 𝑥𝐽) → (𝑥𝐴) = 𝑥)
1211mpteq2dva 5131 . . . . 5 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝑥𝐽 ↦ (𝑥𝐴)) = (𝑥𝐽𝑥))
13 mptresid 5895 . . . . 5 ( I ↾ 𝐽) = (𝑥𝐽𝑥)
1412, 13eqtr4di 2811 . . . 4 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝑥𝐽 ↦ (𝑥𝐴)) = ( I ↾ 𝐽))
1514rneqd 5784 . . 3 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → ran (𝑥𝐽 ↦ (𝑥𝐴)) = ran ( I ↾ 𝐽))
16 rnresi 5920 . . 3 ran ( I ↾ 𝐽) = 𝐽
1715, 16eqtrdi 2809 . 2 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → ran (𝑥𝐽 ↦ (𝑥𝐴)) = 𝐽)
187, 17eqtrd 2793 1 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝐽t 𝐴) = 𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∩ cin 3859   ⊆ wss 3860  𝒫 cpw 4497   ↦ cmpt 5116   I cid 5433  ran crn 5529   ↾ cres 5530  (class class class)co 7156   ↾t crest 16765 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-rest 16767 This theorem is referenced by:  restid  16778  topnid  16780  ssufl  22631
 Copyright terms: Public domain W3C validator