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Theorem resiun2 5956
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 5643 . 2 (𝐶 𝑥𝐴 𝐵) = (𝐶 ∩ ( 𝑥𝐴 𝐵 × V))
2 df-res 5643 . . . . 5 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
32a1i 11 . . . 4 (𝑥𝐴 → (𝐶𝐵) = (𝐶 ∩ (𝐵 × V)))
43iuneq2i 4973 . . 3 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐶 ∩ (𝐵 × V))
5 xpiundir 5701 . . . . 5 ( 𝑥𝐴 𝐵 × V) = 𝑥𝐴 (𝐵 × V)
65ineq2i 4167 . . . 4 (𝐶 ∩ ( 𝑥𝐴 𝐵 × V)) = (𝐶 𝑥𝐴 (𝐵 × V))
7 iunin2 5029 . . . 4 𝑥𝐴 (𝐶 ∩ (𝐵 × V)) = (𝐶 𝑥𝐴 (𝐵 × V))
86, 7eqtr4i 2768 . . 3 (𝐶 ∩ ( 𝑥𝐴 𝐵 × V)) = 𝑥𝐴 (𝐶 ∩ (𝐵 × V))
94, 8eqtr4i 2768 . 2 𝑥𝐴 (𝐶𝐵) = (𝐶 ∩ ( 𝑥𝐴 𝐵 × V))
101, 9eqtr4i 2768 1 (𝐶 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  Vcvv 3443  cin 3907   ciun 4952   × cxp 5629  cres 5633
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-iun 4954  df-opab 5166  df-xp 5637  df-res 5643
This theorem is referenced by:  fvn0ssdmfun  7022  dprd2da  19780
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