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Theorem resindi 5551
Description: Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
Assertion
Ref Expression
resindi (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Proof of Theorem resindi
StepHypRef Expression
1 xpindir 5393 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
21ineq2i 3962 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
3 inindi 3979 . . 3 (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
42, 3eqtri 2793 . 2 (𝐴 ∩ ((𝐵𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
5 df-res 5261 . 2 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
6 df-res 5261 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
7 df-res 5261 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
86, 7ineq12i 3963 . 2 ((𝐴𝐵) ∩ (𝐴𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
94, 5, 83eqtr4i 2803 1 (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  Vcvv 3351  cin 3722   × cxp 5247  cres 5251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-opab 4847  df-xp 5255  df-rel 5256  df-res 5261
This theorem is referenced by:  resindm  5583  gsum2dlem2  18570  fnresin  29763
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