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

Theorem xpiindi 5829
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 5687 . . . . . 6 Rel (𝐶 × 𝐵)
21rgenw 3059 . . . . 5 𝑥𝐴 Rel (𝐶 × 𝐵)
3 r19.2z 4489 . . . . 5 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
42, 3mpan2 688 . . . 4 (𝐴 ≠ ∅ → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
5 reliin 5810 . . . 4 (∃𝑥𝐴 Rel (𝐶 × 𝐵) → Rel 𝑥𝐴 (𝐶 × 𝐵))
64, 5syl 17 . . 3 (𝐴 ≠ ∅ → Rel 𝑥𝐴 (𝐶 × 𝐵))
7 relxp 5687 . . 3 Rel (𝐶 × 𝑥𝐴 𝐵)
86, 7jctil 519 . 2 (𝐴 ≠ ∅ → (Rel (𝐶 × 𝑥𝐴 𝐵) ∧ Rel 𝑥𝐴 (𝐶 × 𝐵)))
9 r19.28zv 4495 . . . . . 6 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝑦𝐶𝑧𝐵) ↔ (𝑦𝐶 ∧ ∀𝑥𝐴 𝑧𝐵)))
109bicomd 222 . . . . 5 (𝐴 ≠ ∅ → ((𝑦𝐶 ∧ ∀𝑥𝐴 𝑧𝐵) ↔ ∀𝑥𝐴 (𝑦𝐶𝑧𝐵)))
11 eliin 4995 . . . . . . 7 (𝑧 ∈ V → (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
1211elv 3474 . . . . . 6 (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵)
1312anbi2i 622 . . . . 5 ((𝑦𝐶𝑧 𝑥𝐴 𝐵) ↔ (𝑦𝐶 ∧ ∀𝑥𝐴 𝑧𝐵))
14 opelxp 5705 . . . . . 6 (⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ (𝑦𝐶𝑧𝐵))
1514ralbii 3087 . . . . 5 (∀𝑥𝐴𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ ∀𝑥𝐴 (𝑦𝐶𝑧𝐵))
1610, 13, 153bitr4g 314 . . . 4 (𝐴 ≠ ∅ → ((𝑦𝐶𝑧 𝑥𝐴 𝐵) ↔ ∀𝑥𝐴𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵)))
17 opelxp 5705 . . . 4 (⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ (𝑦𝐶𝑧 𝑥𝐴 𝐵))
18 opex 5457 . . . . 5 𝑦, 𝑧⟩ ∈ V
19 eliin 4995 . . . . 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 5788 . 2 (((Rel (𝐶 × 𝑥𝐴 𝐵) ∧ Rel 𝑥𝐴 (𝐶 × 𝐵)) ∧ 𝐴 ≠ ∅) → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
238, 22mpancom 685 1 (𝐴 ≠ ∅ → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wne 2934  wral 3055  wrex 3064  Vcvv 3468  c0 4317  cop 4629   ciin 4991   × cxp 5667  Rel wrel 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-iin 4993  df-opab 5204  df-xp 5675  df-rel 5676
This theorem is referenced by:  xpriindi  5830
  Copyright terms: Public domain W3C validator