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Theorem resiun1 4630
 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 3720 . 2 𝑥𝐴 ((𝐶 × V) ∩ 𝐵) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
2 df-res 4357 . . . . 5 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
3 incom 3129 . . . . 5 (𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝐵)
42, 3eqtri 2060 . . . 4 (𝐵𝐶) = ((𝐶 × V) ∩ 𝐵)
54a1i 9 . . 3 (𝑥𝐴 → (𝐵𝐶) = ((𝐶 × V) ∩ 𝐵))
65iuneq2i 3675 . 2 𝑥𝐴 (𝐵𝐶) = 𝑥𝐴 ((𝐶 × V) ∩ 𝐵)
7 df-res 4357 . . 3 ( 𝑥𝐴 𝐵𝐶) = ( 𝑥𝐴 𝐵 ∩ (𝐶 × V))
8 incom 3129 . . 3 ( 𝑥𝐴 𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
97, 8eqtri 2060 . 2 ( 𝑥𝐴 𝐵𝐶) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
101, 6, 93eqtr4ri 2071 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∩ cin 2916  ∪ ciun 3657   × cxp 4343   ↾ cres 4347 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-iun 3659  df-res 4357 This theorem is referenced by: (None)
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