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Theorem xpiindim 4671
Description: Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
Assertion
Ref Expression
xpiindim (∃𝑦 𝑦𝐴 → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐶,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem xpiindim
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4643 . . . . . 6 Rel (𝐶 × 𝐵)
21rgenw 2485 . . . . 5 𝑥𝐴 Rel (𝐶 × 𝐵)
3 r19.2m 3444 . . . . 5 ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
42, 3mpan2 421 . . . 4 (∃𝑦 𝑦𝐴 → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
5 reliin 4656 . . . 4 (∃𝑥𝐴 Rel (𝐶 × 𝐵) → Rel 𝑥𝐴 (𝐶 × 𝐵))
64, 5syl 14 . . 3 (∃𝑦 𝑦𝐴 → Rel 𝑥𝐴 (𝐶 × 𝐵))
7 relxp 4643 . . 3 Rel (𝐶 × 𝑥𝐴 𝐵)
86, 7jctil 310 . 2 (∃𝑦 𝑦𝐴 → (Rel (𝐶 × 𝑥𝐴 𝐵) ∧ Rel 𝑥𝐴 (𝐶 × 𝐵)))
9 eleq1w 2198 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
109cbvexv 1890 . . . . . . 7 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
11 r19.28mv 3450 . . . . . . 7 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝑤𝐶𝑧𝐵) ↔ (𝑤𝐶 ∧ ∀𝑥𝐴 𝑧𝐵)))
1210, 11sylbir 134 . . . . . 6 (∃𝑦 𝑦𝐴 → (∀𝑥𝐴 (𝑤𝐶𝑧𝐵) ↔ (𝑤𝐶 ∧ ∀𝑥𝐴 𝑧𝐵)))
1312bicomd 140 . . . . 5 (∃𝑦 𝑦𝐴 → ((𝑤𝐶 ∧ ∀𝑥𝐴 𝑧𝐵) ↔ ∀𝑥𝐴 (𝑤𝐶𝑧𝐵)))
14 eliin 3813 . . . . . . 7 (𝑧 ∈ V → (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
1514elv 2685 . . . . . 6 (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵)
1615anbi2i 452 . . . . 5 ((𝑤𝐶𝑧 𝑥𝐴 𝐵) ↔ (𝑤𝐶 ∧ ∀𝑥𝐴 𝑧𝐵))
17 opelxp 4564 . . . . . 6 (⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ (𝑤𝐶𝑧𝐵))
1817ralbii 2439 . . . . 5 (∀𝑥𝐴𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ ∀𝑥𝐴 (𝑤𝐶𝑧𝐵))
1913, 16, 183bitr4g 222 . . . 4 (∃𝑦 𝑦𝐴 → ((𝑤𝐶𝑧 𝑥𝐴 𝐵) ↔ ∀𝑥𝐴𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵)))
20 opelxp 4564 . . . 4 (⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ (𝑤𝐶𝑧 𝑥𝐴 𝐵))
21 vex 2684 . . . . . 6 𝑤 ∈ V
22 vex 2684 . . . . . 6 𝑧 ∈ V
2321, 22opex 4146 . . . . 5 𝑤, 𝑧⟩ ∈ V
24 eliin 3813 . . . . 5 (⟨𝑤, 𝑧⟩ ∈ V → (⟨𝑤, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵) ↔ ∀𝑥𝐴𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵)))
2523, 24ax-mp 5 . . . 4 (⟨𝑤, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵) ↔ ∀𝑥𝐴𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵))
2619, 20, 253bitr4g 222 . . 3 (∃𝑦 𝑦𝐴 → (⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵)))
2726eqrelrdv2 4633 . 2 (((Rel (𝐶 × 𝑥𝐴 𝐵) ∧ Rel 𝑥𝐴 (𝐶 × 𝐵)) ∧ ∃𝑦 𝑦𝐴) → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
288, 27mpancom 418 1 (∃𝑦 𝑦𝐴 → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480  wral 2414  wrex 2415  Vcvv 2681  cop 3525   ciin 3809   × cxp 4532  Rel wrel 4539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-iin 3811  df-opab 3985  df-xp 4540  df-rel 4541
This theorem is referenced by:  xpriindim  4672
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