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Theorem resiun2 4659
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 4385 . 2 (𝐶 𝑥𝐴 𝐵) = (𝐶 ∩ ( 𝑥𝐴 𝐵 × V))
2 df-res 4385 . . . . 5 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
32a1i 9 . . . 4 (𝑥𝐴 → (𝐶𝐵) = (𝐶 ∩ (𝐵 × V)))
43iuneq2i 3703 . . 3 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐶 ∩ (𝐵 × V))
5 xpiundir 4427 . . . . 5 ( 𝑥𝐴 𝐵 × V) = 𝑥𝐴 (𝐵 × V)
65ineq2i 3163 . . . 4 (𝐶 ∩ ( 𝑥𝐴 𝐵 × V)) = (𝐶 𝑥𝐴 (𝐵 × V))
7 iunin2 3748 . . . 4 𝑥𝐴 (𝐶 ∩ (𝐵 × V)) = (𝐶 𝑥𝐴 (𝐵 × V))
86, 7eqtr4i 2079 . . 3 (𝐶 ∩ ( 𝑥𝐴 𝐵 × V)) = 𝑥𝐴 (𝐶 ∩ (𝐵 × V))
94, 8eqtr4i 2079 . 2 𝑥𝐴 (𝐶𝐵) = (𝐶 ∩ ( 𝑥𝐴 𝐵 × V))
101, 9eqtr4i 2079 1 (𝐶 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  Vcvv 2574  cin 2944   ciun 3685   × cxp 4371  cres 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-iun 3687  df-opab 3847  df-xp 4379  df-res 4385
This theorem is referenced by: (None)
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