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Theorem inres 4629
 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 3147 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
2 df-res 4357 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 4357 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
43ineq2i 3135 . 2 (𝐴 ∩ (𝐵𝐶)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
51, 2, 43eqtr4ri 2071 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ↾ 𝐶)
 Colors of variables: wff set class Syntax hints:   = wceq 1243  Vcvv 2557   ∩ cin 2916   × cxp 4343   ↾ cres 4347 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-res 4357 This theorem is referenced by: (None)
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