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Theorem resdmdfsn 6051
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 6050 . 2 (Rel 𝑅 → (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (V ∖ {𝑋})))
2 indif1 4288 . . . 4 ((V ∖ {𝑋}) ∩ dom 𝑅) = ((V ∩ dom 𝑅) ∖ {𝑋})
3 incom 4217 . . . . . 6 (V ∩ dom 𝑅) = (dom 𝑅 ∩ V)
4 inv1 4404 . . . . . 6 (dom 𝑅 ∩ V) = dom 𝑅
53, 4eqtri 2763 . . . . 5 (V ∩ dom 𝑅) = dom 𝑅
65difeq1i 4132 . . . 4 ((V ∩ dom 𝑅) ∖ {𝑋}) = (dom 𝑅 ∖ {𝑋})
72, 6eqtri 2763 . . 3 ((V ∖ {𝑋}) ∩ dom 𝑅) = (dom 𝑅 ∖ {𝑋})
87reseq2i 5997 . 2 (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))
91, 8eqtr3di 2790 1 (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  Vcvv 3478  cdif 3960  cin 3962  {csn 4631  dom cdm 5689  cres 5691  Rel wrel 5694
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-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
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-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  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-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-res 5701
This theorem is referenced by:  funresdfunsn  7209
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