Theorem resiun1 5966

Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) (Proof shortened by JJ, 25-Aug-2021.)

Assertion

Ref	Expression
resiun1	⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶)

Distinct variable group:   𝑥,𝐶

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

Proof of Theorem resiun1

Step	Hyp	Ref	Expression
1		iunin1 5029	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ (𝐶 × V)) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V))
2		df-res 5644	. . . 4 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
3	2	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)))
4	3	iuneq2i 4970	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ (𝐶 × V))
5		df-res 5644	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V))
6	1, 4, 5	3eqtr4ri 2771	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∩ cin 3902  ∪ ciun 4948   × cxp 5630   ↾ cres 5634

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-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-in 3910  df-ss 3920  df-iun 4950  df-res 5644

This theorem is referenced by: (None)