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Theorem resiun2 4926
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 4637 . 2 (𝐶 𝑥𝐴 𝐵) = (𝐶 ∩ ( 𝑥𝐴 𝐵 × V))
2 df-res 4637 . . . . 5 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
32a1i 9 . . . 4 (𝑥𝐴 → (𝐶𝐵) = (𝐶 ∩ (𝐵 × V)))
43iuneq2i 3904 . . 3 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐶 ∩ (𝐵 × V))
5 xpiundir 4684 . . . . 5 ( 𝑥𝐴 𝐵 × V) = 𝑥𝐴 (𝐵 × V)
65ineq2i 3333 . . . 4 (𝐶 ∩ ( 𝑥𝐴 𝐵 × V)) = (𝐶 𝑥𝐴 (𝐵 × V))
7 iunin2 3949 . . . 4 𝑥𝐴 (𝐶 ∩ (𝐵 × V)) = (𝐶 𝑥𝐴 (𝐵 × V))
86, 7eqtr4i 2201 . . 3 (𝐶 ∩ ( 𝑥𝐴 𝐵 × V)) = 𝑥𝐴 (𝐶 ∩ (𝐵 × V))
94, 8eqtr4i 2201 . 2 𝑥𝐴 (𝐶𝐵) = (𝐶 ∩ ( 𝑥𝐴 𝐵 × V))
101, 9eqtr4i 2201 1 (𝐶 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wcel 2148  Vcvv 2737  cin 3128   ciun 3886   × cxp 4623  cres 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-iun 3888  df-opab 4064  df-xp 4631  df-res 4637
This theorem is referenced by: (None)
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