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Theorem resindm 4700
Description: When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.)
Assertion
Ref Expression
resindm (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))

Proof of Theorem resindm
StepHypRef Expression
1 resdm 4697 . . 3 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
21ineq2d 3183 . 2 (Rel 𝐴 → ((𝐴𝐵) ∩ (𝐴 ↾ dom 𝐴)) = ((𝐴𝐵) ∩ 𝐴))
3 resindi 4675 . 2 (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ((𝐴𝐵) ∩ (𝐴 ↾ dom 𝐴))
4 incom 3174 . . 3 ((𝐴𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐴𝐵))
5 inres 4677 . . 3 (𝐴 ∩ (𝐴𝐵)) = ((𝐴𝐴) ↾ 𝐵)
6 inidm 3191 . . . 4 (𝐴𝐴) = 𝐴
76reseq1i 4656 . . 3 ((𝐴𝐴) ↾ 𝐵) = (𝐴𝐵)
84, 5, 73eqtrri 2108 . 2 (𝐴𝐵) = ((𝐴𝐵) ∩ 𝐴)
92, 3, 83eqtr4g 2140 1 (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  cin 2981  dom cdm 4391  cres 4393  Rel wrel 4396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-dm 4401  df-res 4403
This theorem is referenced by:  resdmdfsn  4701
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