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Theorem resdmdfsn 4874
 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 indif1 3328 . . . 4 ((V ∖ {𝑋}) ∩ dom 𝑅) = ((V ∩ dom 𝑅) ∖ {𝑋})
2 incom 3275 . . . . . 6 (V ∩ dom 𝑅) = (dom 𝑅 ∩ V)
3 inv1 3406 . . . . . 6 (dom 𝑅 ∩ V) = dom 𝑅
42, 3eqtri 2162 . . . . 5 (V ∩ dom 𝑅) = dom 𝑅
54difeq1i 3197 . . . 4 ((V ∩ dom 𝑅) ∖ {𝑋}) = (dom 𝑅 ∖ {𝑋})
61, 5eqtri 2162 . . 3 ((V ∖ {𝑋}) ∩ dom 𝑅) = (dom 𝑅 ∖ {𝑋})
76reseq2i 4828 . 2 (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))
8 resindm 4873 . 2 (Rel 𝑅 → (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (V ∖ {𝑋})))
97, 8syl5reqr 2189 1 (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332  Vcvv 2691   ∖ cdif 3075   ∩ cin 3077  {csn 3534  dom cdm 4551   ↾ cres 4553  Rel wrel 4556 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1738  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-rab 2427  df-v 2693  df-dif 3080  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-br 3940  df-opab 4000  df-xp 4557  df-rel 4558  df-dm 4561  df-res 4563 This theorem is referenced by:  funresdfunsnss  5635  funresdfunsndc  6414
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