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Theorem xpcogend 15009
Description: The most interesting case of the composition of two Cartesian products. (Contributed by RP, 24-Dec-2019.)
Hypothesis
Ref Expression
xpcogend.1 (𝜑 → (𝐵𝐶) ≠ ∅)
Assertion
Ref Expression
xpcogend (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))

Proof of Theorem xpcogend
Dummy variables 𝑥 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brxp 5737 . . . . . . 7 (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥𝐴𝑦𝐵))
2 brxp 5737 . . . . . . . 8 (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑦𝐶𝑧𝐷))
32biancomi 462 . . . . . . 7 (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑧𝐷𝑦𝐶))
41, 3anbi12i 628 . . . . . 6 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)))
54exbii 1844 . . . . 5 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ∃𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)))
6 an4 656 . . . . . 6 (((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)) ↔ ((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)))
76exbii 1844 . . . . 5 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)) ↔ ∃𝑦((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)))
8 19.42v 1950 . . . . 5 (∃𝑦((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶)))
95, 7, 83bitri 297 . . . 4 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶)))
10 xpcogend.1 . . . . . 6 (𝜑 → (𝐵𝐶) ≠ ∅)
11 ndisj 4375 . . . . . 6 ((𝐵𝐶) ≠ ∅ ↔ ∃𝑦(𝑦𝐵𝑦𝐶))
1210, 11sylib 218 . . . . 5 (𝜑 → ∃𝑦(𝑦𝐵𝑦𝐶))
1312biantrud 531 . . . 4 (𝜑 → ((𝑥𝐴𝑧𝐷) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶))))
149, 13bitr4id 290 . . 3 (𝜑 → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ (𝑥𝐴𝑧𝐷)))
1514opabbidv 5213 . 2 (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐷)})
16 df-co 5697 . 2 ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧)}
17 df-xp 5694 . 2 (𝐴 × 𝐷) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐷)}
1815, 16, 173eqtr4g 2799 1 (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wex 1775  wcel 2105  wne 2937  cin 3961  c0 4338   class class class wbr 5147  {copab 5209   × cxp 5686  ccom 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-co 5697
This theorem is referenced by:  xpcoidgend  15010
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