Theorem xpco 6310

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 4358	. . . . . 6 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
2	1	biimpi 216	. . . . 5 ⊢ (𝐵 ≠ ∅ → ∃𝑦 𝑦 ∈ 𝐵)
3	2	biantrurd 532	. . . 4 ⊢ (𝐵 ≠ ∅ → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑦 𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))))
4		ancom 460	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
5	4	anbi1i 624	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
6		brxp 5737	. . . . . . . 8 ⊢ (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
7		brxp 5737	. . . . . . . 8 ⊢ (𝑦(𝐵 × 𝐶)𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
8	6, 7	anbi12i 628	. . . . . . 7 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
9		anandi 676	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
10	5, 8, 9	3bitr4i 303	. . . . . 6 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
11	10	exbii 1844	. . . . 5 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
12		19.41v 1946	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) ↔ (∃𝑦 𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
13	11, 12	bitr2i 276	. . . 4 ⊢ ((∃𝑦 𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) ↔ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧))
14	3, 13	bitr2di 288	. . 3 ⊢ (𝐵 ≠ ∅ → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
15	14	opabbidv 5213	. 2 ⊢ (𝐵 ≠ ∅ → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)})
16		df-co 5697	. 2 ⊢ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧)}
17		df-xp 5694	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)}
18	15, 16, 17	3eqtr4g 2799	1 ⊢ (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536  ∃wex 1775   ∈ wcel 2105   ≠ wne 2937  ∅c0 4338   class class class wbr 5147  {copab 5209   × cxp 5686   ∘ ccom 5692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-co 5697

This theorem is referenced by:  xpcoid  6311  ustund  24245  ustneism  24247  cosnop  32709