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Theorem xpcogend 14963
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 5731 . . . . . . 7 (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥𝐴𝑦𝐵))
2 brxp 5731 . . . . . . . 8 (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑦𝐶𝑧𝐷))
32biancomi 461 . . . . . . 7 (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑧𝐷𝑦𝐶))
41, 3anbi12i 626 . . . . . 6 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)))
54exbii 1842 . . . . 5 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ∃𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)))
6 an4 654 . . . . . 6 (((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)) ↔ ((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)))
76exbii 1842 . . . . 5 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)) ↔ ∃𝑦((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)))
8 19.42v 1949 . . . . 5 (∃𝑦((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶)))
95, 7, 83bitri 296 . . . 4 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶)))
10 xpcogend.1 . . . . . 6 (𝜑 → (𝐵𝐶) ≠ ∅)
11 ndisj 4371 . . . . . 6 ((𝐵𝐶) ≠ ∅ ↔ ∃𝑦(𝑦𝐵𝑦𝐶))
1210, 11sylib 217 . . . . 5 (𝜑 → ∃𝑦(𝑦𝐵𝑦𝐶))
1312biantrud 530 . . . 4 (𝜑 → ((𝑥𝐴𝑧𝐷) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶))))
149, 13bitr4id 289 . . 3 (𝜑 → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ (𝑥𝐴𝑧𝐷)))
1514opabbidv 5218 . 2 (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐷)})
16 df-co 5691 . 2 ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧)}
17 df-xp 5688 . 2 (𝐴 × 𝐷) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐷)}
1815, 16, 173eqtr4g 2793 1 (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wex 1773  wcel 2098  wne 2937  cin 3948  c0 4326   class class class wbr 5152  {copab 5214   × cxp 5680  ccom 5686
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-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-co 5691
This theorem is referenced by:  xpcoidgend  14964
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