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Theorem xpiindim 4500
 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 4474 . . . . . 6 Rel (𝐶 × 𝐵)
21rgenw 2393 . . . . 5 𝑥𝐴 Rel (𝐶 × 𝐵)
3 eleq1 2116 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
43cbvexv 1811 . . . . . 6 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
5 r19.2m 3336 . . . . . 6 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
64, 5sylanbr 273 . . . . 5 ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
72, 6mpan2 409 . . . 4 (∃𝑦 𝑦𝐴 → ∃𝑥𝐴 Rel (𝐶 × 𝐵))
8 reliin 4486 . . . 4 (∃𝑥𝐴 Rel (𝐶 × 𝐵) → Rel 𝑥𝐴 (𝐶 × 𝐵))
97, 8syl 14 . . 3 (∃𝑦 𝑦𝐴 → Rel 𝑥𝐴 (𝐶 × 𝐵))
10 relxp 4474 . . 3 Rel (𝐶 × 𝑥𝐴 𝐵)
119, 10jctil 299 . 2 (∃𝑦 𝑦𝐴 → (Rel (𝐶 × 𝑥𝐴 𝐵) ∧ Rel 𝑥𝐴 (𝐶 × 𝐵)))
12 r19.28mv 3341 . . . . . . 7 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝑤𝐶𝑧𝐵) ↔ (𝑤𝐶 ∧ ∀𝑥𝐴 𝑧𝐵)))
134, 12sylbir 129 . . . . . 6 (∃𝑦 𝑦𝐴 → (∀𝑥𝐴 (𝑤𝐶𝑧𝐵) ↔ (𝑤𝐶 ∧ ∀𝑥𝐴 𝑧𝐵)))
1413bicomd 133 . . . . 5 (∃𝑦 𝑦𝐴 → ((𝑤𝐶 ∧ ∀𝑥𝐴 𝑧𝐵) ↔ ∀𝑥𝐴 (𝑤𝐶𝑧𝐵)))
15 vex 2577 . . . . . . 7 𝑧 ∈ V
16 eliin 3689 . . . . . . 7 (𝑧 ∈ V → (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
1715, 16ax-mp 7 . . . . . 6 (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵)
1817anbi2i 438 . . . . 5 ((𝑤𝐶𝑧 𝑥𝐴 𝐵) ↔ (𝑤𝐶 ∧ ∀𝑥𝐴 𝑧𝐵))
19 opelxp 4401 . . . . . 6 (⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ (𝑤𝐶𝑧𝐵))
2019ralbii 2347 . . . . 5 (∀𝑥𝐴𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ ∀𝑥𝐴 (𝑤𝐶𝑧𝐵))
2114, 18, 203bitr4g 216 . . . 4 (∃𝑦 𝑦𝐴 → ((𝑤𝐶𝑧 𝑥𝐴 𝐵) ↔ ∀𝑥𝐴𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵)))
22 opelxp 4401 . . . 4 (⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ (𝑤𝐶𝑧 𝑥𝐴 𝐵))
23 vex 2577 . . . . . 6 𝑤 ∈ V
2423, 15opex 3993 . . . . 5 𝑤, 𝑧⟩ ∈ V
25 eliin 3689 . . . . 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 216 . . 3 (∃𝑦 𝑦𝐴 → (⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑥𝐴 (𝐶 × 𝐵)))
2827eqrelrdv2 4466 . 2 (((Rel (𝐶 × 𝑥𝐴 𝐵) ∧ Rel 𝑥𝐴 (𝐶 × 𝐵)) ∧ ∃𝑦 𝑦𝐴) → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
2911, 28mpancom 407 1 (∃𝑦 𝑦𝐴 → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1259  ∃wex 1397   ∈ wcel 1409  ∀wral 2323  ∃wrex 2324  Vcvv 2574  ⟨cop 3405  ∩ ciin 3685   × cxp 4370  Rel wrel 4377 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-iin 3687  df-opab 3846  df-xp 4378  df-rel 4379 This theorem is referenced by:  xpriindim  4501
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