Theorem resundir 5946

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 4214	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2		df-res 5630	. 2 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ∪ 𝐵) ∩ (𝐶 × V))
3		df-res 5630	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5630	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	uneq12i 4096	. 2 ⊢ ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2772	1 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))

Syntax hints:   = wceq 1547  Vcvv 3431   ∪ cun 3881   ∩ cin 3882   × cxp 5616   ↾ cres 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-un 3888  df-in 3890  df-res 5630

This theorem is referenced by:  relresdm1  5985  imaundir  6101  fresaunres2  6699  fvunsn  7123  fvsnun1  7126  fvsnun2  7127  fsnunfv  7131  fsnunres  7132  frrlem12  8237  domss2  9064  axdc3lem4  10366  fseq1p1m1  13543  hashgval  14286  hashinf  14288  setsres  17139  setscom  17141  setsid  17168  pwssplit1  21049  nosupbnd2lem1  27697  noinfbnd2lem1  27712  noetasuplem2  27716  noetasuplem3  27717  noetasuplem4  27718  noetainflem2  27720  ex-res  30529  padct  32810  eulerpartlemt  34555  poimirlem3  37990  ecunres  38761  mapfzcons1  43166  diophrw  43208  eldioph2lem1  43209  eldioph2lem2  43210  diophin  43221  pwssplit4  43534