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Theorem 0rest 16137
 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 4823 . . . 4 ∅ ∈ V
2 restval 16134 . . . 4 ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)))
31, 2mpan 706 . . 3 (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)))
4 mpt0 6059 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ∅
54rneqi 5384 . . . 4 ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ran ∅
6 rn0 5409 . . . 4 ran ∅ = ∅
75, 6eqtri 2673 . . 3 ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ∅
83, 7syl6eq 2701 . 2 (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
9 relxp 5160 . . . 4 Rel (V × V)
10 restfn 16132 . . . . . 6 t Fn (V × V)
11 fndm 6028 . . . . . 6 ( ↾t Fn (V × V) → dom ↾t = (V × V))
1210, 11ax-mp 5 . . . . 5 dom ↾t = (V × V)
1312releqi 5236 . . . 4 (Rel dom ↾t ↔ Rel (V × V))
149, 13mpbir 221 . . 3 Rel dom ↾t
1514ovprc2 6725 . 2 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
168, 15pm2.61i 176 1 (∅ ↾t 𝐴) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∩ cin 3606  ∅c0 3948   ↦ cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  Rel wrel 5148   Fn wfn 5921  (class class class)co 6690   ↾t crest 16128 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-rest 16130 This theorem is referenced by:  firest  16140  topnval  16142  resstopn  21038  ussval  22110  bj-rest00  33159
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