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Theorem resiun1 5574
 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 4737 . 2 𝑥𝐴 (𝐵 ∩ (𝐶 × V)) = ( 𝑥𝐴 𝐵 ∩ (𝐶 × V))
2 df-res 5278 . . . 4 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
32a1i 11 . . 3 (𝑥𝐴 → (𝐵𝐶) = (𝐵 ∩ (𝐶 × V)))
43iuneq2i 4691 . 2 𝑥𝐴 (𝐵𝐶) = 𝑥𝐴 (𝐵 ∩ (𝐶 × V))
5 df-res 5278 . 2 ( 𝑥𝐴 𝐵𝐶) = ( 𝑥𝐴 𝐵 ∩ (𝐶 × V))
61, 4, 53eqtr4ri 2793 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ∩ cin 3714  ∪ ciun 4672   × cxp 5264   ↾ cres 5268 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-iun 4674  df-res 5278 This theorem is referenced by: (None)
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