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Theorem inres 5953
Description: Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
inres (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ↾ 𝐶)

Proof of Theorem inres
StepHypRef Expression
1 inass 4177 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
2 df-res 5643 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 5643 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
43ineq2i 4167 . 2 (𝐴 ∩ (𝐵𝐶)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
51, 2, 43eqtr4ri 2775 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ↾ 𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  Vcvv 3443  cin 3907   × cxp 5629  cres 5633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-in 3915  df-res 5643
This theorem is referenced by:  resindm  5984  fninfp  7116  symgcom2  31777  inres2  36635  xrnres2  36797  br1cossinres  36841
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