MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpiindi Structured version   Visualization version   GIF version

Theorem xpiindi 5489
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 5359 . . . . . 6 Rel (𝐶 × 𝐵)
21rgenw 3132 . . . . 5 𝑥𝐴 Rel (𝐶 × 𝐵)
3 r19.2z 4281 . . . . 5 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
42, 3mpan2 684 . . . 4 (𝐴 ≠ ∅ → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
5 reliin 5474 . . . 4 (∃𝑥𝐴 Rel (𝐶 × 𝐵) → Rel 𝑥𝐴 (𝐶 × 𝐵))
64, 5syl 17 . . 3 (𝐴 ≠ ∅ → Rel 𝑥𝐴 (𝐶 × 𝐵))
7 relxp 5359 . . 3 Rel (𝐶 × 𝑥𝐴 𝐵)
86, 7jctil 517 . 2 (𝐴 ≠ ∅ → (Rel (𝐶 × 𝑥𝐴 𝐵) ∧ Rel 𝑥𝐴 (𝐶 × 𝐵)))
9 r19.28zv 4287 . . . . . 6 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝑦𝐶𝑧𝐵) ↔ (𝑦𝐶 ∧ ∀𝑥𝐴 𝑧𝐵)))
109bicomd 215 . . . . 5 (𝐴 ≠ ∅ → ((𝑦𝐶 ∧ ∀𝑥𝐴 𝑧𝐵) ↔ ∀𝑥𝐴 (𝑦𝐶𝑧𝐵)))
11 vex 3416 . . . . . . 7 𝑧 ∈ V
12 eliin 4744 . . . . . . 7 (𝑧 ∈ V → (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
1311, 12ax-mp 5 . . . . . 6 (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵)
1413anbi2i 618 . . . . 5 ((𝑦𝐶𝑧 𝑥𝐴 𝐵) ↔ (𝑦𝐶 ∧ ∀𝑥𝐴 𝑧𝐵))
15 opelxp 5377 . . . . . 6 (⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ (𝑦𝐶𝑧𝐵))
1615ralbii 3188 . . . . 5 (∀𝑥𝐴𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ ∀𝑥𝐴 (𝑦𝐶𝑧𝐵))
1710, 14, 163bitr4g 306 . . . 4 (𝐴 ≠ ∅ → ((𝑦𝐶𝑧 𝑥𝐴 𝐵) ↔ ∀𝑥𝐴𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵)))
18 opelxp 5377 . . . 4 (⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ (𝑦𝐶𝑧 𝑥𝐴 𝐵))
19 opex 5152 . . . . 5 𝑦, 𝑧⟩ ∈ V
20 eliin 4744 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ V → (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵) ↔ ∀𝑥𝐴𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵)))
2119, 20ax-mp 5 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵) ↔ ∀𝑥𝐴𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵))
2217, 18, 213bitr4g 306 . . 3 (𝐴 ≠ ∅ → (⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵)))
2322eqrelrdv2 5452 . 2 (((Rel (𝐶 × 𝑥𝐴 𝐵) ∧ Rel 𝑥𝐴 (𝐶 × 𝐵)) ∧ 𝐴 ≠ ∅) → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
248, 23mpancom 681 1 (𝐴 ≠ ∅ → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wne 2998  wral 3116  wrex 3117  Vcvv 3413  c0 4143  cop 4402   ciin 4740   × cxp 5339  Rel wrel 5346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-iin 4742  df-opab 4935  df-xp 5347  df-rel 5348
This theorem is referenced by:  xpriindi  5490
  Copyright terms: Public domain W3C validator