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Theorem xpco 6309
Description: Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.)
Assertion
Ref Expression
xpco (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Proof of Theorem xpco
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 4353 . . . . . 6 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
21biimpi 216 . . . . 5 (𝐵 ≠ ∅ → ∃𝑦 𝑦𝐵)
32biantrurd 532 . . . 4 (𝐵 ≠ ∅ → ((𝑥𝐴𝑧𝐶) ↔ (∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶))))
4 ancom 460 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
54anbi1i 624 . . . . . . 7 (((𝑥𝐴𝑦𝐵) ∧ (𝑦𝐵𝑧𝐶)) ↔ ((𝑦𝐵𝑥𝐴) ∧ (𝑦𝐵𝑧𝐶)))
6 brxp 5734 . . . . . . . 8 (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥𝐴𝑦𝐵))
7 brxp 5734 . . . . . . . 8 (𝑦(𝐵 × 𝐶)𝑧 ↔ (𝑦𝐵𝑧𝐶))
86, 7anbi12i 628 . . . . . . 7 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑦𝐵𝑧𝐶)))
9 anandi 676 . . . . . . 7 ((𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ ((𝑦𝐵𝑥𝐴) ∧ (𝑦𝐵𝑧𝐶)))
105, 8, 93bitr4i 303 . . . . . 6 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
1110exbii 1848 . . . . 5 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
12 19.41v 1949 . . . . 5 (∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ (∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
1311, 12bitr2i 276 . . . 4 ((∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧))
143, 13bitr2di 288 . . 3 (𝐵 ≠ ∅ → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑥𝐴𝑧𝐶)))
1514opabbidv 5209 . 2 (𝐵 ≠ ∅ → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐶)})
16 df-co 5694 . 2 ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧)}
17 df-xp 5691 . 2 (𝐴 × 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐶)}
1815, 16, 173eqtr4g 2802 1 (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  c0 4333   class class class wbr 5143  {copab 5205   × cxp 5683  ccom 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-co 5694
This theorem is referenced by:  xpcoid  6310  ustund  24230  ustneism  24232  cosnop  32704
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