Theorem resiun2 6007

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 5670	. 2 ⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V))
2		df-res 5670	. . . . 5 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
3	2	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)))
4	3	iuneq2i 4978	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V))
5		xpiundir 5730	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × V) = ∪ 𝑥 ∈ 𝐴 (𝐵 × V)
6	5	ineq2i 4117	. . . 4 ⊢ (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V)) = (𝐶 ∩ ∪ 𝑥 ∈ 𝐴 (𝐵 × V))
7		iunin2 5034	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V)) = (𝐶 ∩ ∪ 𝑥 ∈ 𝐴 (𝐵 × V))
8	6, 7	eqtr4i 2789	. . 3 ⊢ (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V)) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V))
9	4, 8	eqtr4i 2789	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵) = (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V))
10	1, 9	eqtr4i 2789	1 ⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵)

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092  ∪ ciun 4956   × cxp 5656   ↾ cres 5660

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-un 4154  df-sn 4589  df-pr 4591  df-op 4841  df-iun 4958  df-opab 5174  df-xp 5664  df-res 5670

This theorem is used by:  fvn0ssdmfun  7106  dprd2da  20309