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Theorem xpriindi 5797
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 4976 . . . . . . 7 (𝐴 = ∅ → 𝑥𝐴 𝐵 = 𝑥 ∈ ∅ 𝐵)
2 0iin 5029 . . . . . . 7 𝑥 ∈ ∅ 𝐵 = V
31, 2eqtrdi 2793 . . . . . 6 (𝐴 = ∅ → 𝑥𝐴 𝐵 = V)
43ineq2d 4177 . . . . 5 (𝐴 = ∅ → (𝐷 𝑥𝐴 𝐵) = (𝐷 ∩ V))
5 inv1 4359 . . . . 5 (𝐷 ∩ V) = 𝐷
64, 5eqtrdi 2793 . . . 4 (𝐴 = ∅ → (𝐷 𝑥𝐴 𝐵) = 𝐷)
76xpeq2d 5668 . . 3 (𝐴 = ∅ → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = (𝐶 × 𝐷))
8 iineq1 4976 . . . . . 6 (𝐴 = ∅ → 𝑥𝐴 (𝐶 × 𝐵) = 𝑥 ∈ ∅ (𝐶 × 𝐵))
9 0iin 5029 . . . . . 6 𝑥 ∈ ∅ (𝐶 × 𝐵) = V
108, 9eqtrdi 2793 . . . . 5 (𝐴 = ∅ → 𝑥𝐴 (𝐶 × 𝐵) = V)
1110ineq2d 4177 . . . 4 (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)) = ((𝐶 × 𝐷) ∩ V))
12 inv1 4359 . . . 4 ((𝐶 × 𝐷) ∩ V) = (𝐶 × 𝐷)
1311, 12eqtrdi 2793 . . 3 (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)) = (𝐶 × 𝐷))
147, 13eqtr4d 2780 . 2 (𝐴 = ∅ → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
15 xpindi 5794 . . 3 (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ (𝐶 × 𝑥𝐴 𝐵))
16 xpiindi 5796 . . . 4 (𝐴 ≠ ∅ → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
1716ineq2d 4177 . . 3 (𝐴 ≠ ∅ → ((𝐶 × 𝐷) ∩ (𝐶 × 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
1815, 17eqtrid 2789 . 2 (𝐴 ≠ ∅ → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
1914, 18pm2.61ine 3029 1 (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wne 2944  Vcvv 3448  cin 3914  c0 4287   ciin 4960   × cxp 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-iin 4962  df-opab 5173  df-xp 5644  df-rel 5645
This theorem is referenced by: (None)
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