Theorem xpco 6135

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 4310	. . . . . 6 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
2	1	biimpi 218	. . . . 5 ⊢ (𝐵 ≠ ∅ → ∃𝑦 𝑦 ∈ 𝐵)
3	2	biantrurd 535	. . . 4 ⊢ (𝐵 ≠ ∅ → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑦 𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))))
4		ancom 463	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
5	4	anbi1i 625	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
6		brxp 5596	. . . . . . . 8 ⊢ (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
7		brxp 5596	. . . . . . . 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 674	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
10	5, 8, 9	3bitr4i 305	. . . . . 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 278	. . . 4 ⊢ ((∃𝑦 𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) ↔ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧))
14	3, 13	syl6rbb 290	. . 3 ⊢ (𝐵 ≠ ∅ → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
15	14	opabbidv 5125	. 2 ⊢ (𝐵 ≠ ∅ → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)})
16		df-co 5559	. 2 ⊢ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧)}
17		df-xp 5556	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)}
18	15, 16, 17	3eqtr4g 2881	1 ⊢ (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ≠ wne 3016  ∅c0 4291   class class class wbr 5059  {copab 5121   × cxp 5548   ∘ ccom 5554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-xp 5556  df-co 5559

This theorem is referenced by:  xpcoid  6136  ustund  22824  ustneism  22826  cosnop  30425