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Theorem resiun1 4701
Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
resiun1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem resiun1
StepHypRef Expression
1 iunin2 3778 . 2 𝑥𝐴 ((𝐶 × V) ∩ 𝐵) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
2 df-res 4425 . . . . 5 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
3 incom 3181 . . . . 5 (𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝐵)
42, 3eqtri 2105 . . . 4 (𝐵𝐶) = ((𝐶 × V) ∩ 𝐵)
54a1i 9 . . 3 (𝑥𝐴 → (𝐵𝐶) = ((𝐶 × V) ∩ 𝐵))
65iuneq2i 3733 . 2 𝑥𝐴 (𝐵𝐶) = 𝑥𝐴 ((𝐶 × V) ∩ 𝐵)
7 df-res 4425 . . 3 ( 𝑥𝐴 𝐵𝐶) = ( 𝑥𝐴 𝐵 ∩ (𝐶 × V))
8 incom 3181 . . 3 ( 𝑥𝐴 𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
97, 8eqtri 2105 . 2 ( 𝑥𝐴 𝐵𝐶) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
101, 6, 93eqtr4ri 2116 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Colors of variables: wff set class
Syntax hints:   = wceq 1287  wcel 1436  Vcvv 2615  cin 2987   ciun 3715   × cxp 4411  cres 4415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-in 2994  df-ss 3001  df-iun 3717  df-res 4425
This theorem is referenced by: (None)
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