Theorem resiun2 4649

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 4375	. 2 ⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V))
2		df-res 4375	. . . . 5 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
3	2	a1i 9	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)))
4	3	iuneq2i 3696	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V))
5		xpiundir 4417	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × V) = ∪ 𝑥 ∈ 𝐴 (𝐵 × V)
6	5	ineq2i 3164	. . . 4 ⊢ (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V)) = (𝐶 ∩ ∪ 𝑥 ∈ 𝐴 (𝐵 × V))
7		iunin2 3741	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V)) = (𝐶 ∩ ∪ 𝑥 ∈ 𝐴 (𝐵 × V))
8	6, 7	eqtr4i 2104	. . 3 ⊢ (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V)) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ (𝐵 × V))
9	4, 8	eqtr4i 2104	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵) = (𝐶 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 × V))
10	1, 9	eqtr4i 2104	1 ⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵)

Syntax hints:   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ∩ cin 2972  ∪ ciun 3678   × cxp 4361   ↾ cres 4365

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-iun 3680  df-opab 3840  df-xp 4369  df-res 4375

This theorem is referenced by: (None)