Theorem xpriindim 4870

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 4867	. 2 ⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵))
2		xpiindim 4869	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
3	2	ineq2d 3407	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
4	1, 3	eqtrid 2275	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))

Syntax hints:   → wi 4   = wceq 1397  ∃wex 1540   ∈ wcel 2201   ∩ cin 3198  ∩ ciin 3972   × cxp 4725

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 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-iin 3974  df-opab 4152  df-xp 4733  df-rel 4734

This theorem is referenced by: (None)