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Theorem 0rest 17476
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 5313 . . . 4 ∅ ∈ V
2 restval 17473 . . . 4 ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)))
31, 2mpan 690 . . 3 (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)))
4 mpt0 6711 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ∅
54rneqi 5951 . . . 4 ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ran ∅
6 rn0 5939 . . . 4 ran ∅ = ∅
75, 6eqtri 2763 . . 3 ran (𝑥 ∈ ∅ ↦ (𝑥𝐴)) = ∅
83, 7eqtrdi 2791 . 2 (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
9 relxp 5707 . . . 4 Rel (V × V)
10 restfn 17471 . . . . . 6 t Fn (V × V)
1110fndmi 6673 . . . . 5 dom ↾t = (V × V)
1211releqi 5790 . . . 4 (Rel dom ↾t ↔ Rel (V × V))
139, 12mpbir 231 . . 3 Rel dom ↾t
1413ovprc2 7471 . 2 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
158, 14pm2.61i 182 1 (∅ ↾t 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  c0 4339  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690  Rel wrel 5694  (class class class)co 7431  t crest 17467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-rest 17469
This theorem is referenced by:  firest  17479  topnval  17481  resstopn  23210  ussval  24284  bj-rest00  37064
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