Theorem xpidtr 5119

Description: A square cross product (𝐴 × 𝐴) 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 4750	. . . . . 6 ⊢ (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
2		brxp 4750	. . . . . . . . 9 ⊢ (𝑦(𝐴 × 𝐴)𝑧 ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
3		brxp 4750	. . . . . . . . . . 11 ⊢ (𝑥(𝐴 × 𝐴)𝑧 ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
4	3	simplbi2com 1487	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝑥(𝐴 × 𝐴)𝑧))
5	4	adantl 277	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝑥(𝐴 × 𝐴)𝑧))
6	2, 5	sylbi 121	. . . . . . . 8 ⊢ (𝑦(𝐴 × 𝐴)𝑧 → (𝑥 ∈ 𝐴 → 𝑥(𝐴 × 𝐴)𝑧))
7	6	com12 30	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑦(𝐴 × 𝐴)𝑧 → 𝑥(𝐴 × 𝐴)𝑧))
8	7	adantr 276	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦(𝐴 × 𝐴)𝑧 → 𝑥(𝐴 × 𝐴)𝑧))
9	1, 8	sylbi 121	. . . . 5 ⊢ (𝑥(𝐴 × 𝐴)𝑦 → (𝑦(𝐴 × 𝐴)𝑧 → 𝑥(𝐴 × 𝐴)𝑧))
10	9	imp 124	. . . 4 ⊢ ((𝑥(𝐴 × 𝐴)𝑦 ∧ 𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
11	10	ax-gen 1495	. . 3 ⊢ ∀𝑧((𝑥(𝐴 × 𝐴)𝑦 ∧ 𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
12	11	gen2 1496	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝐴 × 𝐴)𝑦 ∧ 𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
13		cotr 5110	. 2 ⊢ (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) ↔ ∀𝑥∀𝑦∀𝑧((𝑥(𝐴 × 𝐴)𝑦 ∧ 𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧))
14	12, 13	mpbir 146	1 ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1393   ∈ wcel 2200   ⊆ wss 3197   class class class wbr 4083   × cxp 4717   ∘ ccom 4723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-co 4728

This theorem is referenced by:  trinxp  5122  xpider  6753