Theorem xpcogend 15156

Description: The most interesting case of the composition of two Cartesian products. (Contributed by RP, 24-Dec-2019.)

Hypothesis

Ref	Expression
xpcogend.1	⊢ (𝜑 → (𝐵 ∩ 𝐶) ≠ ∅)

Assertion

Ref	Expression
xpcogend	⊢ (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))

Proof of Theorem xpcogend

Dummy variables 𝑥 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brxp 5704	. . . . . . 7 ⊢ (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		brxp 5704	. . . . . . . 8 ⊢ (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷))
3	2	biancomi 468	. . . . . . 7 ⊢ (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶))
4	1, 3	anbi12i 640	. . . . . 6 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶)))
5	4	exbii 1881	. . . . 5 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶)))
6		an4 669	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
7	6	exbii 1881	. . . . 5 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
8		19.42v 1988	. . . . 5 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
9	5, 7, 8	3bitri 300	. . . 4 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
10		xpcogend.1	. . . . . 6 ⊢ (𝜑 → (𝐵 ∩ 𝐶) ≠ ∅)
11		ndisj 4321	. . . . . 6 ⊢ ((𝐵 ∩ 𝐶) ≠ ∅ ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
12	10, 11	sylib 221	. . . . 5 ⊢ (𝜑 → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
13	12	biantrud 541	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
14	9, 13	bitr4id 293	. . 3 ⊢ (𝜑 → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧) ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷)))
15	14	opabbidv 5174	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷)})
16		df-co 5664	. 2 ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧)}
17		df-xp 5661	. 2 ⊢ (𝐴 × 𝐷) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷)}
18	15, 16, 17	3eqtr4g 2823	1 ⊢ (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959   ∩ cin 4092  ∅c0 4281   class class class wbr 5106  {copab 5170   × cxp 5653   ∘ ccom 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-co 5664

This theorem is used by:  xpcoidgend  15157