Theorem resiun1 5056

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 4054	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ((𝐶 × V) ∩ 𝐵) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵)
2		df-res 4760	. . . . 5 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
3		incom 3410	. . . . 5 ⊢ (𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝐵)
4	2, 3	eqtri 2253	. . . 4 ⊢ (𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ 𝐵)
5	4	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ 𝐵))
6	5	iuneq2i 4008	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 ((𝐶 × V) ∩ 𝐵)
7		df-res 4760	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V))
8		incom 3410	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵)
9	7, 8	eqtri 2253	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵)
10	1, 6, 9	3eqtr4ri 2264	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1398   ∈ wcel 2203  Vcvv 2812   ∩ cin 3209  ∪ ciun 3990   × cxp 4746   ↾ cres 4750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-in 3216  df-ss 3223  df-iun 3992  df-res 4760

This theorem is referenced by: (None)