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Theorem xpiindim 4561
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 4535 . . . . . 6 Rel (𝐶 × 𝐵)
21rgenw 2430 . . . . 5 𝑥𝐴 Rel (𝐶 × 𝐵)
3 eleq1 2150 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
43cbvexv 1843 . . . . . 6 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
5 r19.2m 3365 . . . . . 6 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
64, 5sylanbr 279 . . . . 5 ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
72, 6mpan2 416 . . . 4 (∃𝑦 𝑦𝐴 → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
8 reliin 4547 . . . 4 (∃𝑥𝐴 Rel (𝐶 × 𝐵) → Rel 𝑥𝐴 (𝐶 × 𝐵))
97, 8syl 14 . . 3 (∃𝑦 𝑦𝐴 → Rel 𝑥𝐴 (𝐶 × 𝐵))
10 relxp 4535 . . 3 Rel (𝐶 × 𝑥𝐴 𝐵)
119, 10jctil 305 . 2 (∃𝑦 𝑦𝐴 → (Rel (𝐶 × 𝑥𝐴 𝐵) ∧ Rel 𝑥𝐴 (𝐶 × 𝐵)))
12 r19.28mv 3370 . . . . . . 7 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝑤𝐶𝑧𝐵) ↔ (𝑤𝐶 ∧ ∀𝑥𝐴 𝑧𝐵)))
134, 12sylbir 133 . . . . . 6 (∃𝑦 𝑦𝐴 → (∀𝑥𝐴 (𝑤𝐶𝑧𝐵) ↔ (𝑤𝐶 ∧ ∀𝑥𝐴 𝑧𝐵)))
1413bicomd 139 . . . . 5 (∃𝑦 𝑦𝐴 → ((𝑤𝐶 ∧ ∀𝑥𝐴 𝑧𝐵) ↔ ∀𝑥𝐴 (𝑤𝐶𝑧𝐵)))
15 vex 2622 . . . . . . 7 𝑧 ∈ V
16 eliin 3730 . . . . . . 7 (𝑧 ∈ V → (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
1715, 16ax-mp 7 . . . . . 6 (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵)
1817anbi2i 445 . . . . 5 ((𝑤𝐶𝑧 𝑥𝐴 𝐵) ↔ (𝑤𝐶 ∧ ∀𝑥𝐴 𝑧𝐵))
19 opelxp 4457 . . . . . 6 (⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ (𝑤𝐶𝑧𝐵))
2019ralbii 2384 . . . . 5 (∀𝑥𝐴𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ ∀𝑥𝐴 (𝑤𝐶𝑧𝐵))
2114, 18, 203bitr4g 221 . . . 4 (∃𝑦 𝑦𝐴 → ((𝑤𝐶𝑧 𝑥𝐴 𝐵) ↔ ∀𝑥𝐴𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵)))
22 opelxp 4457 . . . 4 (⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ (𝑤𝐶𝑧 𝑥𝐴 𝐵))
23 vex 2622 . . . . . 6 𝑤 ∈ V
2423, 15opex 4047 . . . . 5 𝑤, 𝑧⟩ ∈ V
25 eliin 3730 . . . . 5 (⟨𝑤, 𝑧⟩ ∈ V → (⟨𝑤, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵) ↔ ∀𝑥𝐴𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵)))
2624, 25ax-mp 7 . . . 4 (⟨𝑤, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵) ↔ ∀𝑥𝐴𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵))
2721, 22, 263bitr4g 221 . . 3 (∃𝑦 𝑦𝐴 → (⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵)))
2827eqrelrdv2 4525 . 2 (((Rel (𝐶 × 𝑥𝐴 𝐵) ∧ Rel 𝑥𝐴 (𝐶 × 𝐵)) ∧ ∃𝑦 𝑦𝐴) → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
2911, 28mpancom 413 1 (∃𝑦 𝑦𝐴 → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  wral 2359  wrex 2360  Vcvv 2619  cop 3444   ciin 3726   × cxp 4426  Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-iin 3728  df-opab 3892  df-xp 4434  df-rel 4435
This theorem is referenced by:  xpriindim  4562
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