Theorem xpco 5819

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.

Step	Hyp	Ref	Expression
1		n0 4076	. . . . . 6 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
2	1	biimpi 206	. . . . 5 ⊢ (𝐵 ≠ ∅ → ∃𝑦 𝑦 ∈ 𝐵)
3	2	biantrurd 516	. . . 4 ⊢ (𝐵 ≠ ∅ → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑦 𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))))
4		ancom 452	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
5	4	anbi1i 602	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
6		brxp 5287	. . . . . . . 8 ⊢ (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
7		brxp 5287	. . . . . . . 8 ⊢ (𝑦(𝐵 × 𝐶)𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
8	6, 7	anbi12i 604	. . . . . . 7 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
9		anandi 647	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
10	5, 8, 9	3bitr4i 292	. . . . . 6 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
11	10	exbii 1923	. . . . 5 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
12		19.41v 2028	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) ↔ (∃𝑦 𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
13	11, 12	bitr2i 265	. . . 4 ⊢ ((∃𝑦 𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) ↔ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧))
14	3, 13	syl6rbb 277	. . 3 ⊢ (𝐵 ≠ ∅ → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
15	14	opabbidv 4848	. 2 ⊢ (𝐵 ≠ ∅ → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)})
16		df-co 5258	. 2 ⊢ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧)}
17		df-xp 5255	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)}
18	15, 16, 17	3eqtr4g 2829	1 ⊢ (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630  ∃wex 1851   ∈ wcel 2144   ≠ wne 2942  ∅c0 4061   class class class wbr 4784  {copab 4844   × cxp 5247   ∘ ccom 5253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-co 5258

This theorem is referenced by:  xpcoid  5820  ustund  22244  ustneism  22246