Theorem resundir 5984

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 4231	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2		df-res 5662	. 2 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ∪ 𝐵) ∩ (𝐶 × V))
3		df-res 5662	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5662	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	uneq12i 4165	. 2 ⊢ ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2793	1 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3450   ∩ cin 4089   ∪ cun 4149   × cxp 5648   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-in 4091  df-un 4151  df-res 5662

This theorem is used by:  relresdm1  6024  imaundir  6141  fresaunres2  6754  fvunsn  7184  fvsnun1  7187  fvsnun2  7188  fsnunfv  7192  fsnunres  7193  frrlem12  8315  domss2  9155  axdc3lem4  10531  fseq1p1m1  13732  hashgval  14477  hashinf  14479  setsres  17356  setscom  17358  setsid  17385  pwssplit1  21334  nosupbnd2lem1  28072  noinfbnd2lem1  28087  noetasuplem2  28091  noetasuplem3  28092  noetasuplem4  28093  noetainflem2  28095  ex-res  31042  padct  33310  eulerpartlemt  35003  poimirlem3  38542  ecunres  39327  mapfzcons1  43728  diophrw  43770  eldioph2lem1  43771  eldioph2lem2  43772  diophin  43783  pwssplit4  44091