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Theorem xpriindi 5833
Description: Distributive law for Cartesian product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
xpriindi (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)

Proof of Theorem xpriindi
StepHypRef Expression
1 iineq1 5008 . . . . . . 7 (𝐴 = ∅ → 𝑥𝐴 𝐵 = 𝑥 ∈ ∅ 𝐵)
2 0iin 5061 . . . . . . 7 𝑥 ∈ ∅ 𝐵 = V
31, 2eqtrdi 2783 . . . . . 6 (𝐴 = ∅ → 𝑥𝐴 𝐵 = V)
43ineq2d 4208 . . . . 5 (𝐴 = ∅ → (𝐷 𝑥𝐴 𝐵) = (𝐷 ∩ V))
5 inv1 4390 . . . . 5 (𝐷 ∩ V) = 𝐷
64, 5eqtrdi 2783 . . . 4 (𝐴 = ∅ → (𝐷 𝑥𝐴 𝐵) = 𝐷)
76xpeq2d 5702 . . 3 (𝐴 = ∅ → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = (𝐶 × 𝐷))
8 iineq1 5008 . . . . . 6 (𝐴 = ∅ → 𝑥𝐴 (𝐶 × 𝐵) = 𝑥 ∈ ∅ (𝐶 × 𝐵))
9 0iin 5061 . . . . . 6 𝑥 ∈ ∅ (𝐶 × 𝐵) = V
108, 9eqtrdi 2783 . . . . 5 (𝐴 = ∅ → 𝑥𝐴 (𝐶 × 𝐵) = V)
1110ineq2d 4208 . . . 4 (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)) = ((𝐶 × 𝐷) ∩ V))
12 inv1 4390 . . . 4 ((𝐶 × 𝐷) ∩ V) = (𝐶 × 𝐷)
1311, 12eqtrdi 2783 . . 3 (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)) = (𝐶 × 𝐷))
147, 13eqtr4d 2770 . 2 (𝐴 = ∅ → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
15 xpindi 5830 . . 3 (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ (𝐶 × 𝑥𝐴 𝐵))
16 xpiindi 5832 . . . 4 (𝐴 ≠ ∅ → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
1716ineq2d 4208 . . 3 (𝐴 ≠ ∅ → ((𝐶 × 𝐷) ∩ (𝐶 × 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
1815, 17eqtrid 2779 . 2 (𝐴 ≠ ∅ → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
1914, 18pm2.61ine 3020 1 (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wne 2935  Vcvv 3469  cin 3943  c0 4318   ciin 4992   × cxp 5670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-iin 4994  df-opab 5205  df-xp 5678  df-rel 5679
This theorem is referenced by: (None)
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