Theorem xpcogend 15013

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 5734	. . . . . . 7 ⊢ (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		brxp 5734	. . . . . . . 8 ⊢ (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷))
3	2	biancomi 462	. . . . . . 7 ⊢ (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶))
4	1, 3	anbi12i 628	. . . . . 6 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶)))
5	4	exbii 1848	. . . . 5 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶)))
6		an4 656	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
7	6	exbii 1848	. . . . 5 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
8		19.42v 1953	. . . . 5 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
9	5, 7, 8	3bitri 297	. . . 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 4370	. . . . . 6 ⊢ ((𝐵 ∩ 𝐶) ≠ ∅ ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
12	10, 11	sylib 218	. . . . 5 ⊢ (𝜑 → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
13	12	biantrud 531	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
14	9, 13	bitr4id 290	. . 3 ⊢ (𝜑 → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧) ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷)))
15	14	opabbidv 5209	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷)})
16		df-co 5694	. 2 ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧)}
17		df-xp 5691	. 2 ⊢ (𝐴 × 𝐷) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷)}
18	15, 16, 17	3eqtr4g 2802	1 ⊢ (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940   ∩ cin 3950  ∅c0 4333   class class class wbr 5143  {copab 5205   × cxp 5683   ∘ ccom 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-co 5694

This theorem is referenced by:  xpcoidgend  15014