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Theorem xpidtr 5726
Description: A square Cartesian 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.
StepHypRef Expression
1 brxp 5344 . . . . . 6 (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥𝐴𝑦𝐴))
2 brxp 5344 . . . . . . . . 9 (𝑦(𝐴 × 𝐴)𝑧 ↔ (𝑦𝐴𝑧𝐴))
3 brxp 5344 . . . . . . . . . 10 (𝑥(𝐴 × 𝐴)𝑧 ↔ (𝑥𝐴𝑧𝐴))
43simplbi2com 492 . . . . . . . . 9 (𝑧𝐴 → (𝑥𝐴𝑥(𝐴 × 𝐴)𝑧))
52, 4simplbiim 495 . . . . . . . 8 (𝑦(𝐴 × 𝐴)𝑧 → (𝑥𝐴𝑥(𝐴 × 𝐴)𝑧))
65com12 32 . . . . . . 7 (𝑥𝐴 → (𝑦(𝐴 × 𝐴)𝑧𝑥(𝐴 × 𝐴)𝑧))
76adantr 468 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑦(𝐴 × 𝐴)𝑧𝑥(𝐴 × 𝐴)𝑧))
81, 7sylbi 208 . . . . 5 (𝑥(𝐴 × 𝐴)𝑦 → (𝑦(𝐴 × 𝐴)𝑧𝑥(𝐴 × 𝐴)𝑧))
98imp 395 . . . 4 ((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
109ax-gen 1880 . . 3 𝑧((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
1110gen2 1881 . 2 𝑥𝑦𝑧((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
12 cotr 5715 . 2 (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) ↔ ∀𝑥𝑦𝑧((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧))
1311, 12mpbir 222 1 ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1635  wcel 2158  wss 3766   class class class wbr 4840   × cxp 5306  ccom 5312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pr 5093
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ral 3100  df-rex 3101  df-rab 3104  df-v 3392  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-br 4841  df-opab 4903  df-xp 5314  df-rel 5315  df-co 5317
This theorem is referenced by:  trinxp  5729  xpider  8050  trust  22242  rtrclex  38421  rtrclexi  38425
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