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Theorem bj-xpcossxp 35660
Description: The composition of two Cartesian products is included in the expected Cartesian product. There is equality if (𝐵𝐶) ≠ ∅, see xpcogend 14859. (Contributed by BJ, 22-May-2024.)
Assertion
Ref Expression
bj-xpcossxp ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷)

Proof of Theorem bj-xpcossxp
Dummy variables 𝑥 𝑦 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brxp 5681 . . . . . . 7 (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥𝐴𝑦𝐵))
2 brxp 5681 . . . . . . 7 (𝑦(𝐶 × 𝐷)𝑡 ↔ (𝑦𝐶𝑡𝐷))
31, 2anbi12i 627 . . . . . 6 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑡) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑦𝐶𝑡𝐷)))
4 an43 656 . . . . . 6 (((𝑥𝐴𝑦𝐵) ∧ (𝑦𝐶𝑡𝐷)) ↔ ((𝑥𝐴𝑡𝐷) ∧ (𝑦𝐵𝑦𝐶)))
53, 4bitri 274 . . . . 5 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑡) ↔ ((𝑥𝐴𝑡𝐷) ∧ (𝑦𝐵𝑦𝐶)))
65exbii 1850 . . . 4 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑡) ↔ ∃𝑦((𝑥𝐴𝑡𝐷) ∧ (𝑦𝐵𝑦𝐶)))
7 19.42v 1957 . . . . 5 (∃𝑦((𝑥𝐴𝑡𝐷) ∧ (𝑦𝐵𝑦𝐶)) ↔ ((𝑥𝐴𝑡𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶)))
87simplbi 498 . . . 4 (∃𝑦((𝑥𝐴𝑡𝐷) ∧ (𝑦𝐵𝑦𝐶)) → (𝑥𝐴𝑡𝐷))
96, 8sylbi 216 . . 3 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑡) → (𝑥𝐴𝑡𝐷))
109ssopab2i 5507 . 2 {⟨𝑥, 𝑡⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑡)} ⊆ {⟨𝑥, 𝑡⟩ ∣ (𝑥𝐴𝑡𝐷)}
11 df-co 5642 . 2 ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑡⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑡)}
12 df-xp 5639 . 2 (𝐴 × 𝐷) = {⟨𝑥, 𝑡⟩ ∣ (𝑥𝐴𝑡𝐷)}
1310, 11, 123sstr4i 3987 1 ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wex 1781  wcel 2106  wss 3910   class class class wbr 5105  {copab 5167   × cxp 5631  ccom 5637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-co 5642
This theorem is referenced by:  bj-imdirco  35661
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