MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0rest Structured version   Visualization version   GIF version

Theorem 0rest 16691
Description: Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
0rest (∅ ↾t 𝐴) = ∅

Proof of Theorem 0rest
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 0ex 5202 . . . 4 ∅ ∈ V
2 restval 16688 . . . 4 ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)))
31, 2mpan 686 . . 3 (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)))
4 mpt0 6483 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ∅
54rneqi 5800 . . . 4 ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ran ∅
6 rn0 5789 . . . 4 ran ∅ = ∅
75, 6eqtri 2841 . . 3 ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ∅
83, 7syl6eq 2869 . 2 (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
9 relxp 5566 . . . 4 Rel (V × V)
10 restfn 16686 . . . . . 6 t Fn (V × V)
11 fndm 6448 . . . . . 6 ( ↾t Fn (V × V) → dom ↾t = (V × V))
1210, 11ax-mp 5 . . . . 5 dom ↾t = (V × V)
1312releqi 5645 . . . 4 (Rel dom ↾t ↔ Rel (V × V))
149, 13mpbir 232 . . 3 Rel dom ↾t
1514ovprc2 7185 . 2 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
168, 15pm2.61i 183 1 (∅ ↾t 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  c0 4288  cmpt 5137   × cxp 5546  dom cdm 5548  ran crn 5549  Rel wrel 5553   Fn wfn 6343  (class class class)co 7145  t crest 16682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-rest 16684
This theorem is referenced by:  firest  16694  topnval  16696  resstopn  21722  ussval  22795  bj-rest00  34266
  Copyright terms: Public domain W3C validator