Theorem xpriindim 4868

Description: Distributive law for cross product over relativized indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)

Assertion

Ref	Expression
xpriindim	⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐶,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem xpriindim

Step	Hyp	Ref	Expression
1		xpindi 4865	. 2 ⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵))
2		xpiindim 4867	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
3	2	ineq2d 3408	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
4	1, 3	eqtrid 2276	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))

Syntax hints:   → wi 4   = wceq 1397  ∃wex 1540   ∈ wcel 2202   ∩ cin 3199  ∩ ciin 3971   × cxp 4723

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-iin 3973  df-opab 4151  df-xp 4731  df-rel 4732

This theorem is referenced by: (None)