Theorem bj-xpcossxp 38204

Description: The composition of two Cartesian products is included in the expected Cartesian product. There is equality if (𝐵 ∩ 𝐶) ≠ ∅, see xpcogend 15133. (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 5699	. . . . . . 7 ⊢ (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		brxp 5699	. . . . . . 7 ⊢ (𝑦(𝐶 × 𝐷)𝑡 ↔ (𝑦 ∈ 𝐶 ∧ 𝑡 ∈ 𝐷))
3	1, 2	anbi12i 640	. . . . . 6 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑡 ∈ 𝐷)))
4		an43 671	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑡 ∈ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
5	3, 4	bitri 278	. . . . 5 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
6	5	exbii 1880	. . . 4 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
7		19.42v 1985	. . . . 5 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
8	7	simplbi 502	. . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷))
9	6, 8	sylbi 220	. . 3 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) → (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷))
10	9	ssopab2i 5524	. 2 ⊢ {⟨𝑥, 𝑡⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡)} ⊆ {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷)}
11		df-co 5659	. 2 ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑡⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡)}
12		df-xp 5656	. 2 ⊢ (𝐴 × 𝐷) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷)}
13	10, 11, 12	3sstr4i 3959	1 ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷)

This proof depends on syntax axioms:   ∧ wa 401  ∃wex 1811   ∈ wcel 2144   ⊆ wss 3895   class class class wbr 5102  {copab 5166   × cxp 5648   ∘ ccom 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-co 5659

This theorem is used by:  bj-imdirco  38205