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Theorem inres 5864
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 4193 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
2 df-res 5560 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 5560 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
43ineq2i 4183 . 2 (𝐴 ∩ (𝐵𝐶)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
51, 2, 43eqtr4ri 2852 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ↾ 𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  Vcvv 3492  cin 3932   × cxp 5546  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-in 3940  df-res 5560
This theorem is referenced by:  resindm  5893  fninfp  6928  symgcom2  30655  inres2  35387  xrnres2  35531  br1cossinres  35567
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