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Theorem 0rest 16762
 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 5178 . . . 4 ∅ ∈ V
2 restval 16759 . . . 4 ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)))
31, 2mpan 690 . . 3 (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)))
4 mpt0 6474 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ∅
54rneqi 5779 . . . 4 ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ran ∅
6 rn0 5768 . . . 4 ran ∅ = ∅
75, 6eqtri 2782 . . 3 ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ∅
83, 7eqtrdi 2810 . 2 (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
9 relxp 5543 . . . 4 Rel (V × V)
10 restfn 16757 . . . . . 6 t Fn (V × V)
1110fndmi 6438 . . . . 5 dom ↾t = (V × V)
1211releqi 5622 . . . 4 (Rel dom ↾t ↔ Rel (V × V))
139, 12mpbir 234 . . 3 Rel dom ↾t
1413ovprc2 7191 . 2 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
158, 14pm2.61i 185 1 (∅ ↾t 𝐴) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1539   ∈ wcel 2112  Vcvv 3410   ∩ cin 3858  ∅c0 4226   ↦ cmpt 5113   × cxp 5523  dom cdm 5525  ran crn 5526  Rel wrel 5530  (class class class)co 7151   ↾t crest 16753 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7694  df-2nd 7695  df-rest 16755 This theorem is referenced by:  firest  16765  topnval  16767  resstopn  21887  ussval  22961  bj-rest00  34777
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