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Theorem resiun1 4573
 Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
resiun1 ( x A B𝐶) = x A (B𝐶)
Distinct variable group:   x,𝐶
Allowed substitution hints:   A(x)   B(x)

Proof of Theorem resiun1
StepHypRef Expression
1 iunin2 3711 . 2 x A ((𝐶 × V) ∩ B) = ((𝐶 × V) ∩ x A B)
2 df-res 4300 . . . . 5 (B𝐶) = (B ∩ (𝐶 × V))
3 incom 3123 . . . . 5 (B ∩ (𝐶 × V)) = ((𝐶 × V) ∩ B)
42, 3eqtri 2057 . . . 4 (B𝐶) = ((𝐶 × V) ∩ B)
54a1i 9 . . 3 (x A → (B𝐶) = ((𝐶 × V) ∩ B))
65iuneq2i 3666 . 2 x A (B𝐶) = x A ((𝐶 × V) ∩ B)
7 df-res 4300 . . 3 ( x A B𝐶) = ( x A B ∩ (𝐶 × V))
8 incom 3123 . . 3 ( x A B ∩ (𝐶 × V)) = ((𝐶 × V) ∩ x A B)
97, 8eqtri 2057 . 2 ( x A B𝐶) = ((𝐶 × V) ∩ x A B)
101, 6, 93eqtr4ri 2068 1 ( x A B𝐶) = x A (B𝐶)
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∩ cin 2910  ∪ ciun 3648   × cxp 4286   ↾ cres 4290 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iun 3650  df-res 4300 This theorem is referenced by: (None)
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