Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpco Structured version   Visualization version   GIF version

Theorem xpco 6109
 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 4260 . . . . . 6 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
21biimpi 219 . . . . 5 (𝐵 ≠ ∅ → ∃𝑦 𝑦𝐵)
32biantrurd 536 . . . 4 (𝐵 ≠ ∅ → ((𝑥𝐴𝑧𝐶) ↔ (∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶))))
4 ancom 464 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
54anbi1i 626 . . . . . . 7 (((𝑥𝐴𝑦𝐵) ∧ (𝑦𝐵𝑧𝐶)) ↔ ((𝑦𝐵𝑥𝐴) ∧ (𝑦𝐵𝑧𝐶)))
6 brxp 5566 . . . . . . . 8 (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥𝐴𝑦𝐵))
7 brxp 5566 . . . . . . . 8 (𝑦(𝐵 × 𝐶)𝑧 ↔ (𝑦𝐵𝑧𝐶))
86, 7anbi12i 629 . . . . . . 7 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑦𝐵𝑧𝐶)))
9 anandi 675 . . . . . . 7 ((𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ ((𝑦𝐵𝑥𝐴) ∧ (𝑦𝐵𝑧𝐶)))
105, 8, 93bitr4i 306 . . . . . 6 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
1110exbii 1849 . . . . 5 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
12 19.41v 1950 . . . . 5 (∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ (∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
1311, 12bitr2i 279 . . . 4 ((∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧))
143, 13syl6rbb 291 . . 3 (𝐵 ≠ ∅ → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑥𝐴𝑧𝐶)))
1514opabbidv 5097 . 2 (𝐵 ≠ ∅ → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐶)})
16 df-co 5529 . 2 ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧)}
17 df-xp 5526 . 2 (𝐴 × 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐶)}
1815, 16, 173eqtr4g 2858 1 (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∅c0 4243   class class class wbr 5031  {copab 5093   × cxp 5518   ∘ ccom 5524 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-xp 5526  df-co 5529 This theorem is referenced by:  xpcoid  6110  ustund  22837  ustneism  22839  cosnop  30465
 Copyright terms: Public domain W3C validator