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Theorem resiun1 5866
Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) (Proof shortened by JJ, 25-Aug-2021.)
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 iunin1 4985 . 2 𝑥𝐴 (𝐵 ∩ (𝐶 × V)) = ( 𝑥𝐴 𝐵 ∩ (𝐶 × V))
2 df-res 5560 . . . 4 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
32a1i 11 . . 3 (𝑥𝐴 → (𝐵𝐶) = (𝐵 ∩ (𝐶 × V)))
43iuneq2i 4931 . 2 𝑥𝐴 (𝐵𝐶) = 𝑥𝐴 (𝐵 ∩ (𝐶 × V))
5 df-res 5560 . 2 ( 𝑥𝐴 𝐵𝐶) = ( 𝑥𝐴 𝐵 ∩ (𝐶 × V))
61, 4, 53eqtr4ri 2852 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932   ciun 4910   × cxp 5546  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-in 3940  df-ss 3949  df-iun 4912  df-res 5560
This theorem is referenced by: (None)
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