Theorem resiun2 5965

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 4955	. . 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 4157	. . . 4 ⊢ (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V)) = (𝐶 ∩ ∪ 𝑥 ∈ 𝐴 (𝐵 × V))
7		iunin2 5013	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V)) = (𝐶 ∩ ∪ 𝑥 ∈ 𝐴 (𝐵 × V))
8	6, 7	eqtr4i 2762	. . 3 ⊢ (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V)) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V))
9	4, 8	eqtr4i 2762	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵) = (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V))
10	1, 9	eqtr4i 2762	1 ⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵)

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∩ cin 3888  ∪ ciun 4933   × cxp 5629   ↾ cres 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-un 3894  df-in 3896  df-ss 3906  df-sn 4568  df-pr 4570  df-op 4574  df-iun 4935  df-opab 5148  df-xp 5637  df-res 5643

This theorem is referenced by:  fvn0ssdmfun  7026  dprd2da  20019