Theorem resiun1 5952

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 5022	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ (𝐶 × V)) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V))
2		df-res 5631	. . . 4 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
3	2	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)))
4	3	iuneq2i 4963	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ (𝐶 × V))
5		df-res 5631	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V))
6	1, 4, 5	3eqtr4ri 2767	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∩ cin 3897  ∪ ciun 4941   × cxp 5617   ↾ cres 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-in 3905  df-ss 3915  df-iun 4943  df-res 5631

This theorem is referenced by: (None)