Theorem bj-xpcossxp 37390

Description: The composition of two Cartesian products is included in the expected Cartesian product. There is equality if (𝐵 ∩ 𝐶) ≠ ∅, see xpcogend 14897. (Contributed by BJ, 22-May-2024.)

Assertion

Ref	Expression
bj-xpcossxp	⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷)

Proof of Theorem bj-xpcossxp

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

Step	Hyp	Ref	Expression
1		brxp 5673	. . . . . . 7 ⊢ (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		brxp 5673	. . . . . . 7 ⊢ (𝑦(𝐶 × 𝐷)𝑡 ↔ (𝑦 ∈ 𝐶 ∧ 𝑡 ∈ 𝐷))
3	1, 2	anbi12i 628	. . . . . 6 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑡 ∈ 𝐷)))
4		an43 658	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑡 ∈ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
5	3, 4	bitri 275	. . . . 5 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
6	5	exbii 1849	. . . 4 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
7		19.42v 1954	. . . . 5 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
8	7	simplbi 497	. . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷))
9	6, 8	sylbi 217	. . 3 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) → (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷))
10	9	ssopab2i 5498	. 2 ⊢ {⟨𝑥, 𝑡⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡)} ⊆ {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷)}
11		df-co 5633	. 2 ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑡⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡)}
12		df-xp 5630	. 2 ⊢ (𝐴 × 𝐷) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷)}
13	10, 11, 12	3sstr4i 3985	1 ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷)

Syntax hints:   ∧ wa 395  ∃wex 1780   ∈ wcel 2113   ⊆ wss 3901   class class class wbr 5098  {copab 5160   × cxp 5622   ∘ ccom 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-co 5633

This theorem is referenced by:  bj-imdirco  37391