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Theorem xpidtr 6123
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 5725 . . . . . 6 (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥𝐴𝑦𝐴))
2 brxp 5725 . . . . . . . . 9 (𝑦(𝐴 × 𝐴)𝑧 ↔ (𝑦𝐴𝑧𝐴))
3 brxp 5725 . . . . . . . . . 10 (𝑥(𝐴 × 𝐴)𝑧 ↔ (𝑥𝐴𝑧𝐴))
43simplbi2com 503 . . . . . . . . 9 (𝑧𝐴 → (𝑥𝐴𝑥(𝐴 × 𝐴)𝑧))
52, 4simplbiim 505 . . . . . . . 8 (𝑦(𝐴 × 𝐴)𝑧 → (𝑥𝐴𝑥(𝐴 × 𝐴)𝑧))
65com12 32 . . . . . . 7 (𝑥𝐴 → (𝑦(𝐴 × 𝐴)𝑧𝑥(𝐴 × 𝐴)𝑧))
76adantr 481 . . . . . 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 407 . . . 4 ((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
109ax-gen 1797 . . 3 𝑧((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
1110gen2 1798 . 2 𝑥𝑦𝑧((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
12 cotr 6111 . 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 396  wal 1539  wcel 2106  wss 3948   class class class wbr 5148   × cxp 5674  ccom 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-co 5685
This theorem is referenced by:  trinxp  6126  xpider  8781  trust  23733  rtrclex  42358  rtrclexi  42362
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