Theorem xpriindim 4820

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 4817	. 2 ⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵))
2		xpiindim 4819	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
3	2	ineq2d 3375	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
4	1, 3	eqtrid 2251	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))

Syntax hints:   → wi 4   = wceq 1373  ∃wex 1516   ∈ wcel 2177   ∩ cin 3166  ∩ ciin 3930   × cxp 4677

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-iin 3932  df-opab 4110  df-xp 4685  df-rel 4686

This theorem is referenced by: (None)