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

Theorem xpco 6290
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 4346 . . . . . 6 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
21biimpi 215 . . . . 5 (𝐵 ≠ ∅ → ∃𝑦 𝑦𝐵)
32biantrurd 531 . . . 4 (𝐵 ≠ ∅ → ((𝑥𝐴𝑧𝐶) ↔ (∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶))))
4 ancom 459 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
54anbi1i 622 . . . . . . 7 (((𝑥𝐴𝑦𝐵) ∧ (𝑦𝐵𝑧𝐶)) ↔ ((𝑦𝐵𝑥𝐴) ∧ (𝑦𝐵𝑧𝐶)))
6 brxp 5721 . . . . . . . 8 (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥𝐴𝑦𝐵))
7 brxp 5721 . . . . . . . 8 (𝑦(𝐵 × 𝐶)𝑧 ↔ (𝑦𝐵𝑧𝐶))
86, 7anbi12i 626 . . . . . . 7 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑦𝐵𝑧𝐶)))
9 anandi 674 . . . . . . 7 ((𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ ((𝑦𝐵𝑥𝐴) ∧ (𝑦𝐵𝑧𝐶)))
105, 8, 93bitr4i 302 . . . . . 6 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
1110exbii 1843 . . . . 5 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
12 19.41v 1946 . . . . 5 (∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ (∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
1311, 12bitr2i 275 . . . 4 ((∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧))
143, 13bitr2di 287 . . 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 5681 . 2 ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧)}
17 df-xp 5678 . 2 (𝐴 × 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐶)}
1815, 16, 173eqtr4g 2791 1 (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wex 1774  wcel 2099  wne 2930  c0 4322   class class class wbr 5143  {copab 5205   × cxp 5670  ccom 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5144  df-opab 5206  df-xp 5678  df-co 5681
This theorem is referenced by:  xpcoid  6291  ustund  24211  ustneism  24213  cosnop  32604
  Copyright terms: Public domain W3C validator