Theorem resiun2 5960

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 5643	. 2 ⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V))
2		df-res 5643	. . . . 5 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
3	2	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)))
4	3	iuneq2i 4973	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V))
5		xpiundir 5703	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × V) = ∪ 𝑥 ∈ 𝐴 (𝐵 × V)
6	5	ineq2i 4176	. . . 4 ⊢ (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V)) = (𝐶 ∩ ∪ 𝑥 ∈ 𝐴 (𝐵 × V))
7		iunin2 5030	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V)) = (𝐶 ∩ ∪ 𝑥 ∈ 𝐴 (𝐵 × V))
8	6, 7	eqtr4i 2755	. . 3 ⊢ (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V)) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V))
9	4, 8	eqtr4i 2755	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵) = (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V))
10	1, 9	eqtr4i 2755	1 ⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵)

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∩ cin 3910  ∪ ciun 4951   × cxp 5629   ↾ cres 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-iun 4953  df-opab 5165  df-xp 5637  df-res 5643

This theorem is referenced by:  fvn0ssdmfun  7028  dprd2da  19958