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Theorem resiun1 6020
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 5077 . 2 𝑥𝐴 (𝐵 ∩ (𝐶 × V)) = ( 𝑥𝐴 𝐵 ∩ (𝐶 × V))
2 df-res 5701 . . . 4 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
32a1i 11 . . 3 (𝑥𝐴 → (𝐵𝐶) = (𝐵 ∩ (𝐶 × V)))
43iuneq2i 5018 . 2 𝑥𝐴 (𝐵𝐶) = 𝑥𝐴 (𝐵 ∩ (𝐶 × V))
5 df-res 5701 . 2 ( 𝑥𝐴 𝐵𝐶) = ( 𝑥𝐴 𝐵 ∩ (𝐶 × V))
61, 4, 53eqtr4ri 2774 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962   ciun 4996   × cxp 5687  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-iun 4998  df-res 5701
This theorem is referenced by: (None)
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