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Theorem resindm 5364
 Description: Class restriction distributes over intersection. (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 5361 . . 3 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
21ineq2d 3776 . 2 (Rel 𝐴 → ((𝐴𝐵) ∩ (𝐴 ↾ dom 𝐴)) = ((𝐴𝐵) ∩ 𝐴))
3 resindi 5332 . 2 (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ((𝐴𝐵) ∩ (𝐴 ↾ dom 𝐴))
4 incom 3767 . . 3 ((𝐴𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐴𝐵))
5 inres 5334 . . 3 (𝐴 ∩ (𝐴𝐵)) = ((𝐴𝐴) ↾ 𝐵)
6 inidm 3784 . . . 4 (𝐴𝐴) = 𝐴
76reseq1i 5313 . . 3 ((𝐴𝐴) ↾ 𝐵) = (𝐴𝐵)
84, 5, 73eqtrri 2637 . 2 (𝐴𝐵) = ((𝐴𝐵) ∩ 𝐴)
92, 3, 83eqtr4g 2669 1 (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∩ cin 3539  dom cdm 5038   ↾ cres 5040  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048  df-res 5050 This theorem is referenced by:  resdmdfsn  5365  resfifsupp  8186  poimirlem3  32582  fresin2  38347  cncfuni  38772  fourierdlem48  39047  fourierdlem49  39048  fourierdlem113  39112  sssmf  39625
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