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Theorem xpco 5634
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 3907 . . . . . 6 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
21biimpi 206 . . . . 5 (𝐵 ≠ ∅ → ∃𝑦 𝑦𝐵)
32biantrurd 529 . . . 4 (𝐵 ≠ ∅ → ((𝑥𝐴𝑧𝐶) ↔ (∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶))))
4 ancom 466 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
54anbi1i 730 . . . . . . 7 (((𝑥𝐴𝑦𝐵) ∧ (𝑦𝐵𝑧𝐶)) ↔ ((𝑦𝐵𝑥𝐴) ∧ (𝑦𝐵𝑧𝐶)))
6 brxp 5107 . . . . . . . 8 (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥𝐴𝑦𝐵))
7 brxp 5107 . . . . . . . 8 (𝑦(𝐵 × 𝐶)𝑧 ↔ (𝑦𝐵𝑧𝐶))
86, 7anbi12i 732 . . . . . . 7 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑦𝐵𝑧𝐶)))
9 anandi 870 . . . . . . 7 ((𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ ((𝑦𝐵𝑥𝐴) ∧ (𝑦𝐵𝑧𝐶)))
105, 8, 93bitr4i 292 . . . . . 6 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
1110exbii 1771 . . . . 5 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
12 19.41v 1911 . . . . 5 (∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ (∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)))
1311, 12bitr2i 265 . . . 4 ((∃𝑦 𝑦𝐵 ∧ (𝑥𝐴𝑧𝐶)) ↔ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧))
143, 13syl6rbb 277 . . 3 (𝐵 ≠ ∅ → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑥𝐴𝑧𝐶)))
1514opabbidv 4678 . 2 (𝐵 ≠ ∅ → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐶)})
16 df-co 5083 . 2 ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐵 × 𝐶)𝑧)}
17 df-xp 5080 . 2 (𝐴 × 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐶)}
1815, 16, 173eqtr4g 2680 1 (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  c0 3891   class class class wbr 4613  {copab 4672   × cxp 5072  ccom 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-co 5083
This theorem is referenced by:  xpcoid  5635  ustund  21935  ustneism  21937
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