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Theorem inres 5714
 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 4078 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
2 df-res 5416 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 5416 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
43ineq2i 4068 . 2 (𝐴 ∩ (𝐵𝐶)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
51, 2, 43eqtr4ri 2808 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ↾ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1508  Vcvv 3410   ∩ cin 3823   × cxp 5402   ↾ cres 5406 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-rab 3092  df-v 3412  df-in 3831  df-res 5416 This theorem is referenced by:  resindm  5743  fninfp  6758  inres2  34983  xrnres2  35129  br1cossinres  35165
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