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Theorem inres 4806
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 3256 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
2 df-res 4521 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 4521 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
43ineq2i 3244 . 2 (𝐴 ∩ (𝐵𝐶)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
51, 2, 43eqtr4ri 2149 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ↾ 𝐶)
Colors of variables: wff set class
Syntax hints:   = wceq 1316  Vcvv 2660  cin 3040   × cxp 4507  cres 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-res 4521
This theorem is referenced by:  resindm  4831
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