Theorem resundir 5995

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 4239	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2		df-res 5675	. 2 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ∪ 𝐵) ∩ (𝐶 × V))
3		df-res 5675	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5675	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	uneq12i 4120	. 2 ⊢ ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2798	1 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3457   ∪ cun 3904   ∩ cin 3905   × cxp 5661   ↾ cres 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-un 3911  df-in 3913  df-res 5675

This theorem is used by:  relresdm1  6037  imaundir  6150  fresaunres2  6754  fvunsn  7183  fvsnun1  7186  fvsnun2  7187  fsnunfv  7191  fsnunres  7192  frrlem12  8300  domss2  9131  axdc3lem4  10452  fseq1p1m1  13645  hashgval  14389  hashinf  14391  setsres  17262  setscom  17264  setsid  17291  pwssplit1  21232  nosupbnd2lem1  27932  noinfbnd2lem1  27947  noetasuplem2  27951  noetasuplem3  27952  noetasuplem4  27953  noetainflem2  27955  ex-res  30865  padct  33135  eulerpartlemt  34828  poimirlem3  38333  ecunres  39103  mapfzcons1  43508  diophrw  43550  eldioph2lem1  43551  eldioph2lem2  43552  diophin  43563  pwssplit4  43876