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Theorem xpidtr 4743
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.
StepHypRef Expression
1 brxp 4403 . . . . . 6 (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥𝐴𝑦𝐴))
2 brxp 4403 . . . . . . . . 9 (𝑦(𝐴 × 𝐴)𝑧 ↔ (𝑦𝐴𝑧𝐴))
3 brxp 4403 . . . . . . . . . . 11 (𝑥(𝐴 × 𝐴)𝑧 ↔ (𝑥𝐴𝑧𝐴))
43simplbi2com 1349 . . . . . . . . . 10 (𝑧𝐴 → (𝑥𝐴𝑥(𝐴 × 𝐴)𝑧))
54adantl 266 . . . . . . . . 9 ((𝑦𝐴𝑧𝐴) → (𝑥𝐴𝑥(𝐴 × 𝐴)𝑧))
62, 5sylbi 118 . . . . . . . 8 (𝑦(𝐴 × 𝐴)𝑧 → (𝑥𝐴𝑥(𝐴 × 𝐴)𝑧))
76com12 30 . . . . . . 7 (𝑥𝐴 → (𝑦(𝐴 × 𝐴)𝑧𝑥(𝐴 × 𝐴)𝑧))
87adantr 265 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑦(𝐴 × 𝐴)𝑧𝑥(𝐴 × 𝐴)𝑧))
91, 8sylbi 118 . . . . 5 (𝑥(𝐴 × 𝐴)𝑦 → (𝑦(𝐴 × 𝐴)𝑧𝑥(𝐴 × 𝐴)𝑧))
109imp 119 . . . 4 ((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
1110ax-gen 1354 . . 3 𝑧((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
1211gen2 1355 . 2 𝑥𝑦𝑧((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧)
13 cotr 4734 . 2 (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) ↔ ∀𝑥𝑦𝑧((𝑥(𝐴 × 𝐴)𝑦𝑦(𝐴 × 𝐴)𝑧) → 𝑥(𝐴 × 𝐴)𝑧))
1412, 13mpbir 138 1 ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  wcel 1409  wss 2945   class class class wbr 3792   × cxp 4371  ccom 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-co 4382
This theorem is referenced by:  trinxp  4746  xpiderm  6208
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