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Theorem resiun2 5848
 Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
resiun2 (𝐶 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶𝐵)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem resiun2
StepHypRef Expression
1 df-res 5539 . 2 (𝐶 𝑥𝐴 𝐵) = (𝐶 ∩ ( 𝑥𝐴 𝐵 × V))
2 df-res 5539 . . . . 5 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
32a1i 11 . . . 4 (𝑥𝐴 → (𝐶𝐵) = (𝐶 ∩ (𝐵 × V)))
43iuneq2i 4907 . . 3 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐶 ∩ (𝐵 × V))
5 xpiundir 5596 . . . . 5 ( 𝑥𝐴 𝐵 × V) = 𝑥𝐴 (𝐵 × V)
65ineq2i 4116 . . . 4 (𝐶 ∩ ( 𝑥𝐴 𝐵 × V)) = (𝐶 𝑥𝐴 (𝐵 × V))
7 iunin2 4961 . . . 4 𝑥𝐴 (𝐶 ∩ (𝐵 × V)) = (𝐶 𝑥𝐴 (𝐵 × V))
86, 7eqtr4i 2784 . . 3 (𝐶 ∩ ( 𝑥𝐴 𝐵 × V)) = 𝑥𝐴 (𝐶 ∩ (𝐵 × V))
94, 8eqtr4i 2784 . 2 𝑥𝐴 (𝐶𝐵) = (𝐶 ∩ ( 𝑥𝐴 𝐵 × V))
101, 9eqtr4i 2784 1 (𝐶 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∩ cin 3859  ∪ ciun 4886   × cxp 5525   ↾ cres 5529 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-iun 4888  df-opab 5098  df-xp 5533  df-res 5539 This theorem is referenced by:  fvn0ssdmfun  6838  dprd2da  19237
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