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Theorem xpiindi 5796
Description: Distributive law for Cartesian product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
xpiindi (𝐴 ≠ ∅ → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem xpiindi
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 5656 . . . . . 6 Rel (𝐶 × 𝐵)
21rgenw 3069 . . . . 5 𝑥𝐴 Rel (𝐶 × 𝐵)
3 r19.2z 4457 . . . . 5 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
42, 3mpan2 690 . . . 4 (𝐴 ≠ ∅ → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
5 reliin 5778 . . . 4 (∃𝑥𝐴 Rel (𝐶 × 𝐵) → Rel 𝑥𝐴 (𝐶 × 𝐵))
64, 5syl 17 . . 3 (𝐴 ≠ ∅ → Rel 𝑥𝐴 (𝐶 × 𝐵))
7 relxp 5656 . . 3 Rel (𝐶 × 𝑥𝐴 𝐵)
86, 7jctil 521 . 2 (𝐴 ≠ ∅ → (Rel (𝐶 × 𝑥𝐴 𝐵) ∧ Rel 𝑥𝐴 (𝐶 × 𝐵)))
9 r19.28zv 4463 . . . . . 6 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝑦𝐶𝑧𝐵) ↔ (𝑦𝐶 ∧ ∀𝑥𝐴 𝑧𝐵)))
109bicomd 222 . . . . 5 (𝐴 ≠ ∅ → ((𝑦𝐶 ∧ ∀𝑥𝐴 𝑧𝐵) ↔ ∀𝑥𝐴 (𝑦𝐶𝑧𝐵)))
11 eliin 4964 . . . . . . 7 (𝑧 ∈ V → (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
1211elv 3454 . . . . . 6 (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵)
1312anbi2i 624 . . . . 5 ((𝑦𝐶𝑧 𝑥𝐴 𝐵) ↔ (𝑦𝐶 ∧ ∀𝑥𝐴 𝑧𝐵))
14 opelxp 5674 . . . . . 6 (⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ (𝑦𝐶𝑧𝐵))
1514ralbii 3097 . . . . 5 (∀𝑥𝐴𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ ∀𝑥𝐴 (𝑦𝐶𝑧𝐵))
1610, 13, 153bitr4g 314 . . . 4 (𝐴 ≠ ∅ → ((𝑦𝐶𝑧 𝑥𝐴 𝐵) ↔ ∀𝑥𝐴𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵)))
17 opelxp 5674 . . . 4 (⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ (𝑦𝐶𝑧 𝑥𝐴 𝐵))
18 opex 5426 . . . . 5 𝑦, 𝑧⟩ ∈ V
19 eliin 4964 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ V → (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵) ↔ ∀𝑥𝐴𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵)))
2018, 19ax-mp 5 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵) ↔ ∀𝑥𝐴𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵))
2116, 17, 203bitr4g 314 . . 3 (𝐴 ≠ ∅ → (⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵)))
2221eqrelrdv2 5756 . 2 (((Rel (𝐶 × 𝑥𝐴 𝐵) ∧ Rel 𝑥𝐴 (𝐶 × 𝐵)) ∧ 𝐴 ≠ ∅) → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
238, 22mpancom 687 1 (𝐴 ≠ ∅ → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  Vcvv 3448  c0 4287  cop 4597   ciin 4960   × cxp 5636  Rel wrel 5643
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-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:  xpriindi  5797
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