Theorem resiun1 6003

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 5041	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ (𝐶 × V)) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V))
2		df-res 5678	. . . 4 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
3	2	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)))
4	3	iuneq2i 4983	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ (𝐶 × V))
5		df-res 5678	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V))
6	1, 4, 5	3eqtr4ri 2800	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶)

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  Vcvv 3458   ∩ cin 3907  ∪ ciun 4961   × cxp 5664   ↾ cres 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-in 3915  df-ss 3925  df-iun 4963  df-res 5678

This theorem is used by: (None)