Theorem resiun2 4879

Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
resiun2	⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵)

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem resiun2

Step	Hyp	Ref	Expression
1		df-res 4591	. 2 ⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V))
2		df-res 4591	. . . . 5 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
3	2	a1i 9	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)))
4	3	iuneq2i 3863	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V))
5		xpiundir 4638	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × V) = ∪ 𝑥 ∈ 𝐴 (𝐵 × V)
6	5	ineq2i 3301	. . . 4 ⊢ (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V)) = (𝐶 ∩ ∪ 𝑥 ∈ 𝐴 (𝐵 × V))
7		iunin2 3908	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V)) = (𝐶 ∩ ∪ 𝑥 ∈ 𝐴 (𝐵 × V))
8	6, 7	eqtr4i 2178	. . 3 ⊢ (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V)) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V))
9	4, 8	eqtr4i 2178	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵) = (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V))
10	1, 9	eqtr4i 2178	1 ⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵)

Syntax hints:   = wceq 1332   ∈ wcel 2125  Vcvv 2709   ∩ cin 3097  ∪ ciun 3845   × cxp 4577   ↾ cres 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-iun 3847  df-opab 4022  df-xp 4585  df-res 4591

This theorem is referenced by: (None)