Theorem xpidtr 6116

Description: A Cartesian square is a transitive relation. (Contributed by FL, 31-Jul-2009.)

Assertion

Ref	Expression
xpidtr	⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)

Proof of Theorem xpidtr

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

Step	Hyp	Ref	Expression
1		brxp 5708	. . . . . 6 ⊢ (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
2		brxp 5708	. . . . . . . . 9 ⊢ (𝑦(𝐴 × 𝐴)𝑧 ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
3		brxp 5708	. . . . . . . . . 10 ⊢ (𝑥(𝐴 × 𝐴)𝑧 ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
4	3	simplbi2com 502	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝑥(𝐴 × 𝐴)𝑧))
5	2, 4	simplbiim 504	. . . . . . . 8 ⊢ (𝑦(𝐴 × 𝐴)𝑧 → (𝑥 ∈ 𝐴 → 𝑥(𝐴 × 𝐴)𝑧))
6	5	com12 32	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑦(𝐴 × 𝐴)𝑧 → 𝑥(𝐴 × 𝐴)𝑧))
7	6	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦(𝐴 × 𝐴)𝑧 → 𝑥(𝐴 × 𝐴)𝑧))
8	1, 7	sylbi 217	. . . . 5 ⊢ (𝑥(𝐴 × 𝐴)𝑦 → (𝑦(𝐴 × 𝐴)𝑧 → 𝑥(𝐴 × 𝐴)𝑧))
9	8	imp 406	. . . 4 ⊢ ((𝑥(𝐴 × 𝐴)𝑦 ∧ 𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
10	9	ax-gen 1795	. . 3 ⊢ ∀𝑧((𝑥(𝐴 × 𝐴)𝑦 ∧ 𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
11	10	gen2 1796	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝐴 × 𝐴)𝑦 ∧ 𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
12		cotr 6104	. 2 ⊢ (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) ↔ ∀𝑥∀𝑦∀𝑧((𝑥(𝐴 × 𝐴)𝑦 ∧ 𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧))
13	11, 12	mpbir 231	1 ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   ∈ wcel 2109   ⊆ wss 3931   class class class wbr 5124   × cxp 5657   ∘ ccom 5663

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-co 5668

This theorem is referenced by:  trinxp  6119  xpider  8807  trust  24173  rtrclex  43608  rtrclexi  43612