ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  resiun1 GIF version

Theorem resiun1 4910
Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
resiun1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem resiun1
StepHypRef Expression
1 iunin2 3936 . 2 𝑥𝐴 ((𝐶 × V) ∩ 𝐵) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
2 df-res 4623 . . . . 5 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
3 incom 3319 . . . . 5 (𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝐵)
42, 3eqtri 2191 . . . 4 (𝐵𝐶) = ((𝐶 × V) ∩ 𝐵)
54a1i 9 . . 3 (𝑥𝐴 → (𝐵𝐶) = ((𝐶 × V) ∩ 𝐵))
65iuneq2i 3891 . 2 𝑥𝐴 (𝐵𝐶) = 𝑥𝐴 ((𝐶 × V) ∩ 𝐵)
7 df-res 4623 . . 3 ( 𝑥𝐴 𝐵𝐶) = ( 𝑥𝐴 𝐵 ∩ (𝐶 × V))
8 incom 3319 . . 3 ( 𝑥𝐴 𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
97, 8eqtri 2191 . 2 ( 𝑥𝐴 𝐵𝐶) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
101, 6, 93eqtr4ri 2202 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Colors of variables: wff set class
Syntax hints:   = wceq 1348  wcel 2141  Vcvv 2730  cin 3120   ciun 3873   × cxp 4609  cres 4613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-iun 3875  df-res 4623
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator