Theorem xpidtr 5155

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 4782	. . . . . 6 ⊢ (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
2		brxp 4782	. . . . . . . . 9 ⊢ (𝑦(𝐴 × 𝐴)𝑧 ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
3		brxp 4782	. . . . . . . . . . 11 ⊢ (𝑥(𝐴 × 𝐴)𝑧 ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
4	3	simplbi2com 1490	. . . . . . . . . 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 1498	. . 3 ⊢ ∀𝑧((𝑥(𝐴 × 𝐴)𝑦 ∧ 𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
12	11	gen2 1499	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝐴 × 𝐴)𝑦 ∧ 𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
13		cotr 5146	. 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 1396   ∈ wcel 2205   ⊆ wss 3213   class class class wbr 4111   × cxp 4749   ∘ ccom 4755

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-co 4760

This theorem is referenced by:  trinxp  5158  xpider  6842