Theorem resundir 6004

Description: Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)

Assertion

Ref	Expression
resundir	⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))

Proof of Theorem resundir

Step	Hyp	Ref	Expression
1		indir 4277	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2		df-res 5694	. 2 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ∪ 𝐵) ∩ (𝐶 × V))
3		df-res 5694	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5694	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	uneq12i 4161	. 2 ⊢ ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2764	1 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))

Syntax hints:   = wceq 1534  Vcvv 3462   ∪ cun 3945   ∩ cin 3946   × cxp 5680   ↾ cres 5684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-un 3952  df-in 3954  df-res 5694

This theorem is referenced by:  relresdm1  6042  imaundir  6162  fresaunres2  6774  fvunsn  7193  fvsnun1  7196  fvsnun2  7197  fsnunfv  7201  fsnunres  7202  frrlem12  8312  wfrlem14OLD  8352  domss2  9174  axdc3lem4  10496  fseq1p1m1  13629  hashgval  14350  hashinf  14352  setsres  17180  setscom  17182  setsid  17210  pwssplit1  21037  nosupbnd2lem1  27745  noinfbnd2lem1  27760  noetasuplem2  27764  noetasuplem3  27765  noetasuplem4  27766  noetainflem2  27768  ex-res  30374  padct  32633  eulerpartlemt  34205  poimirlem3  37324  mapfzcons1  42374  diophrw  42416  eldioph2lem1  42417  eldioph2lem2  42418  diophin  42429  pwssplit4  42750