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Theorem xpidtr 4658
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 4318 . . . . . 6 (x(A × A)y ↔ (x A y A))
2 brxp 4318 . . . . . . . . 9 (y(A × A)z ↔ (y A z A))
3 brxp 4318 . . . . . . . . . . 11 (x(A × A)z ↔ (x A z A))
43simplbi2com 1330 . . . . . . . . . 10 (z A → (x Ax(A × A)z))
54adantl 262 . . . . . . . . 9 ((y A z A) → (x Ax(A × A)z))
62, 5sylbi 114 . . . . . . . 8 (y(A × A)z → (x Ax(A × A)z))
76com12 27 . . . . . . 7 (x A → (y(A × A)zx(A × A)z))
87adantr 261 . . . . . 6 ((x A y A) → (y(A × A)zx(A × A)z))
91, 8sylbi 114 . . . . 5 (x(A × A)y → (y(A × A)zx(A × A)z))
109imp 115 . . . 4 ((x(A × A)y y(A × A)z) → x(A × A)z)
1110ax-gen 1335 . . 3 z((x(A × A)y y(A × A)z) → x(A × A)z)
1211gen2 1336 . 2 xyz((x(A × A)y y(A × A)z) → x(A × A)z)
13 cotr 4649 . 2 (((A × A) ∘ (A × A)) ⊆ (A × A) ↔ xyz((x(A × A)y y(A × A)z) → x(A × A)z))
1412, 13mpbir 134 1 ((A × A) ∘ (A × A)) ⊆ (A × A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   wcel 1390  wss 2911   class class class wbr 3755   × cxp 4286  ccom 4292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-co 4297
This theorem is referenced by:  trinxp  4661  xpiderm  6113
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