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Theorem resdmdfsn 6030
Description: Restricting a relation to its domain without a set is the same as restricting the relation to the universe without this set. (Contributed by AV, 2-Dec-2018.)
Assertion
Ref Expression
resdmdfsn (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))

Proof of Theorem resdmdfsn
StepHypRef Expression
1 resindm 6029 . 2 (Rel 𝑅 → (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (V ∖ {𝑋})))
2 indif1 4266 . . . 4 ((V ∖ {𝑋}) ∩ dom 𝑅) = ((V ∩ dom 𝑅) ∖ {𝑋})
3 incom 4195 . . . . . 6 (V ∩ dom 𝑅) = (dom 𝑅 ∩ V)
4 inv1 4390 . . . . . 6 (dom 𝑅 ∩ V) = dom 𝑅
53, 4eqtri 2753 . . . . 5 (V ∩ dom 𝑅) = dom 𝑅
65difeq1i 4110 . . . 4 ((V ∩ dom 𝑅) ∖ {𝑋}) = (dom 𝑅 ∖ {𝑋})
72, 6eqtri 2753 . . 3 ((V ∖ {𝑋}) ∩ dom 𝑅) = (dom 𝑅 ∖ {𝑋})
87reseq2i 5976 . 2 (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))
91, 8eqtr3di 2780 1 (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  Vcvv 3463  cdif 3936  cin 3938  {csn 4624  dom cdm 5672  cres 5674  Rel wrel 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-dm 5682  df-res 5684
This theorem is referenced by:  funresdfunsn  7194
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