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Theorem xpco 6242
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 4307 . . . . . 6 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
21biimpi 215 . . . . 5 (𝐵 ≠ ∅ → ∃𝑦 𝑦𝐵)
32biantrurd 534 . . . 4 (𝐵 ≠ ∅ → ((𝑥𝐴𝑧𝐶) ↔ (∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶))))
4 ancom 462 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
54anbi1i 625 . . . . . . 7 (((𝑥𝐴𝑦𝐵) ∧ (𝑦𝐵𝑧𝐶)) ↔ ((𝑦𝐵𝑥𝐴) ∧ (𝑦𝐵𝑧𝐶)))
6 brxp 5682 . . . . . . . 8 (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥𝐴𝑦𝐵))
7 brxp 5682 . . . . . . . 8 (𝑦(𝐵 × 𝐶)𝑧 ↔ (𝑦𝐵𝑧𝐶))
86, 7anbi12i 628 . . . . . . 7 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑦𝐵𝑧𝐶)))
9 anandi 675 . . . . . . 7 ((𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ ((𝑦𝐵𝑥𝐴) ∧ (𝑦𝐵𝑧𝐶)))
105, 8, 93bitr4i 303 . . . . . 6 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
1110exbii 1851 . . . . 5 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
12 19.41v 1954 . . . . 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 5172 . 2 (𝐵 ≠ ∅ → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐶)})
16 df-co 5643 . 2 ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧)}
17 df-xp 5640 . 2 (𝐴 × 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐶)}
1815, 16, 173eqtr4g 2798 1 (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2940  c0 4283   class class class wbr 5106  {copab 5168   × cxp 5632  ccom 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-co 5643
This theorem is referenced by:  xpcoid  6243  ustund  23589  ustneism  23591  cosnop  31656
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