Theorem resiun1 4678

Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

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		iunin2 3761	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ((𝐶 × V) ∩ 𝐵) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵)
2		df-res 4403	. . . . 5 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
3		incom 3174	. . . . 5 ⊢ (𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝐵)
4	2, 3	eqtri 2103	. . . 4 ⊢ (𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ 𝐵)
5	4	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ 𝐵))
6	5	iuneq2i 3716	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 ((𝐶 × V) ∩ 𝐵)
7		df-res 4403	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V))
8		incom 3174	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵)
9	7, 8	eqtri 2103	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵)
10	1, 6, 9	3eqtr4ri 2114	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1285   ∈ wcel 1434  Vcvv 2610   ∩ cin 2981  ∪ ciun 3698   × cxp 4389   ↾ cres 4393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-in 2988  df-ss 2995  df-iun 3700  df-res 4403

This theorem is referenced by: (None)