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Theorem resiun1 4887
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 3914 . 2 𝑥𝐴 ((𝐶 × V) ∩ 𝐵) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
2 df-res 4600 . . . . 5 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
3 incom 3300 . . . . 5 (𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝐵)
42, 3eqtri 2178 . . . 4 (𝐵𝐶) = ((𝐶 × V) ∩ 𝐵)
54a1i 9 . . 3 (𝑥𝐴 → (𝐵𝐶) = ((𝐶 × V) ∩ 𝐵))
65iuneq2i 3869 . 2 𝑥𝐴 (𝐵𝐶) = 𝑥𝐴 ((𝐶 × V) ∩ 𝐵)
7 df-res 4600 . . 3 ( 𝑥𝐴 𝐵𝐶) = ( 𝑥𝐴 𝐵 ∩ (𝐶 × V))
8 incom 3300 . . 3 ( 𝑥𝐴 𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
97, 8eqtri 2178 . 2 ( 𝑥𝐴 𝐵𝐶) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
101, 6, 93eqtr4ri 2189 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Colors of variables: wff set class
Syntax hints:   = wceq 1335  wcel 2128  Vcvv 2712  cin 3101   ciun 3851   × cxp 4586  cres 4590
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-iun 3853  df-res 4600
This theorem is referenced by: (None)
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