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Theorem resiun2 5558
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 5262 . 2 (𝐶 𝑥𝐴 𝐵) = (𝐶 ∩ ( 𝑥𝐴 𝐵 × V))
2 df-res 5262 . . . . 5 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
32a1i 11 . . . 4 (𝑥𝐴 → (𝐶𝐵) = (𝐶 ∩ (𝐵 × V)))
43iuneq2i 4674 . . 3 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐶 ∩ (𝐵 × V))
5 xpiundir 5313 . . . . 5 ( 𝑥𝐴 𝐵 × V) = 𝑥𝐴 (𝐵 × V)
65ineq2i 3962 . . . 4 (𝐶 ∩ ( 𝑥𝐴 𝐵 × V)) = (𝐶 𝑥𝐴 (𝐵 × V))
7 iunin2 4719 . . . 4 𝑥𝐴 (𝐶 ∩ (𝐵 × V)) = (𝐶 𝑥𝐴 (𝐵 × V))
86, 7eqtr4i 2796 . . 3 (𝐶 ∩ ( 𝑥𝐴 𝐵 × V)) = 𝑥𝐴 (𝐶 ∩ (𝐵 × V))
94, 8eqtr4i 2796 . 2 𝑥𝐴 (𝐶𝐵) = (𝐶 ∩ ( 𝑥𝐴 𝐵 × V))
101, 9eqtr4i 2796 1 (𝐶 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  Vcvv 3351  cin 3722   ciun 4655   × cxp 5248  cres 5252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-iun 4657  df-opab 4848  df-xp 5256  df-res 5262
This theorem is referenced by:  fvn0ssdmfun  6495  dprd2da  18649
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