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Theorem 0rest 17357
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 5300 . . . 4 ∅ ∈ V
2 restval 17354 . . . 4 ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)))
31, 2mpan 688 . . 3 (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)))
4 mpt0 6679 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ∅
54rneqi 5928 . . . 4 ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ran ∅
6 rn0 5917 . . . 4 ran ∅ = ∅
75, 6eqtri 2759 . . 3 ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ∅
83, 7eqtrdi 2787 . 2 (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
9 relxp 5687 . . . 4 Rel (V × V)
10 restfn 17352 . . . . . 6 t Fn (V × V)
1110fndmi 6642 . . . . 5 dom ↾t = (V × V)
1211releqi 5769 . . . 4 (Rel dom ↾t ↔ Rel (V × V))
139, 12mpbir 230 . . 3 Rel dom ↾t
1413ovprc2 7433 . 2 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
158, 14pm2.61i 182 1 (∅ ↾t 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  Vcvv 3473  cin 3943  c0 4318  cmpt 5224   × cxp 5667  dom cdm 5669  ran crn 5670  Rel wrel 5674  (class class class)co 7393  t crest 17348
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-rest 17350
This theorem is referenced by:  firest  17360  topnval  17362  resstopn  22619  ussval  23693  bj-rest00  35766
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