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Theorem xpidtr 6114
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.
StepHypRef Expression
1 brxp 5716 . . . . . 6 (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥𝐴𝑦𝐴))
2 brxp 5716 . . . . . . . . 9 (𝑦(𝐴 × 𝐴)𝑧 ↔ (𝑦𝐴𝑧𝐴))
3 brxp 5716 . . . . . . . . . 10 (𝑥(𝐴 × 𝐴)𝑧 ↔ (𝑥𝐴𝑧𝐴))
43simplbi2com 502 . . . . . . . . 9 (𝑧𝐴 → (𝑥𝐴𝑥(𝐴 × 𝐴)𝑧))
52, 4simplbiim 504 . . . . . . . 8 (𝑦(𝐴 × 𝐴)𝑧 → (𝑥𝐴𝑥(𝐴 × 𝐴)𝑧))
65com12 32 . . . . . . 7 (𝑥𝐴 → (𝑦(𝐴 × 𝐴)𝑧𝑥(𝐴 × 𝐴)𝑧))
76adantr 480 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑦(𝐴 × 𝐴)𝑧𝑥(𝐴 × 𝐴)𝑧))
81, 7sylbi 216 . . . . 5 (𝑥(𝐴 × 𝐴)𝑦 → (𝑦(𝐴 × 𝐴)𝑧𝑥(𝐴 × 𝐴)𝑧))
98imp 406 . . . 4 ((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
109ax-gen 1789 . . 3 𝑧((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
1110gen2 1790 . 2 𝑥𝑦𝑧((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
12 cotr 6102 . 2 (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) ↔ ∀𝑥𝑦𝑧((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧))
1311, 12mpbir 230 1 ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1531  wcel 2098  wss 3941   class class class wbr 5139   × cxp 5665  ccom 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-co 5676
This theorem is referenced by:  trinxp  6117  xpider  8779  trust  24058  rtrclex  42882  rtrclexi  42886
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