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Theorem xpider 6468
Description: A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
xpider  |-  ( A  X.  A )  Er  A

Proof of Theorem xpider
StepHypRef Expression
1 relxp 4618 . 2  |-  Rel  ( A  X.  A )
2 dmxpid 4730 . 2  |-  dom  ( A  X.  A )  =  A
3 cnvxp 4927 . . 3  |-  `' ( A  X.  A )  =  ( A  X.  A )
4 xpidtr 4899 . . 3  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  C_  ( A  X.  A
)
5 uneq1 3193 . . . 4  |-  ( `' ( A  X.  A
)  =  ( A  X.  A )  -> 
( `' ( A  X.  A )  u.  ( A  X.  A
) )  =  ( ( A  X.  A
)  u.  ( A  X.  A ) ) )
6 unss2 3217 . . . 4  |-  ( ( ( A  X.  A
)  o.  ( A  X.  A ) ) 
C_  ( A  X.  A )  ->  ( `' ( A  X.  A )  u.  (
( A  X.  A
)  o.  ( A  X.  A ) ) )  C_  ( `' ( A  X.  A
)  u.  ( A  X.  A ) ) )
7 unidm 3189 . . . . 5  |-  ( ( A  X.  A )  u.  ( A  X.  A ) )  =  ( A  X.  A
)
8 eqtr 2135 . . . . . 6  |-  ( ( ( `' ( A  X.  A )  u.  ( A  X.  A
) )  =  ( ( A  X.  A
)  u.  ( A  X.  A ) )  /\  ( ( A  X.  A )  u.  ( A  X.  A
) )  =  ( A  X.  A ) )  ->  ( `' ( A  X.  A
)  u.  ( A  X.  A ) )  =  ( A  X.  A ) )
9 sseq2 3091 . . . . . . 7  |-  ( ( `' ( A  X.  A )  u.  ( A  X.  A ) )  =  ( A  X.  A )  ->  (
( `' ( A  X.  A )  u.  ( ( A  X.  A )  o.  ( A  X.  A ) ) )  C_  ( `' ( A  X.  A
)  u.  ( A  X.  A ) )  <-> 
( `' ( A  X.  A )  u.  ( ( A  X.  A )  o.  ( A  X.  A ) ) )  C_  ( A  X.  A ) ) )
109biimpd 143 . . . . . 6  |-  ( ( `' ( A  X.  A )  u.  ( A  X.  A ) )  =  ( A  X.  A )  ->  (
( `' ( A  X.  A )  u.  ( ( A  X.  A )  o.  ( A  X.  A ) ) )  C_  ( `' ( A  X.  A
)  u.  ( A  X.  A ) )  ->  ( `' ( A  X.  A )  u.  ( ( A  X.  A )  o.  ( A  X.  A
) ) )  C_  ( A  X.  A
) ) )
118, 10syl 14 . . . . 5  |-  ( ( ( `' ( A  X.  A )  u.  ( A  X.  A
) )  =  ( ( A  X.  A
)  u.  ( A  X.  A ) )  /\  ( ( A  X.  A )  u.  ( A  X.  A
) )  =  ( A  X.  A ) )  ->  ( ( `' ( A  X.  A )  u.  (
( A  X.  A
)  o.  ( A  X.  A ) ) )  C_  ( `' ( A  X.  A
)  u.  ( A  X.  A ) )  ->  ( `' ( A  X.  A )  u.  ( ( A  X.  A )  o.  ( A  X.  A
) ) )  C_  ( A  X.  A
) ) )
127, 11mpan2 421 . . . 4  |-  ( ( `' ( A  X.  A )  u.  ( A  X.  A ) )  =  ( ( A  X.  A )  u.  ( A  X.  A
) )  ->  (
( `' ( A  X.  A )  u.  ( ( A  X.  A )  o.  ( A  X.  A ) ) )  C_  ( `' ( A  X.  A
)  u.  ( A  X.  A ) )  ->  ( `' ( A  X.  A )  u.  ( ( A  X.  A )  o.  ( A  X.  A
) ) )  C_  ( A  X.  A
) ) )
135, 6, 12syl2im 38 . . 3  |-  ( `' ( A  X.  A
)  =  ( A  X.  A )  -> 
( ( ( A  X.  A )  o.  ( A  X.  A
) )  C_  ( A  X.  A )  -> 
( `' ( A  X.  A )  u.  ( ( A  X.  A )  o.  ( A  X.  A ) ) )  C_  ( A  X.  A ) ) )
143, 4, 13mp2 16 . 2  |-  ( `' ( A  X.  A
)  u.  ( ( A  X.  A )  o.  ( A  X.  A ) ) ) 
C_  ( A  X.  A )
15 df-er 6397 . 2  |-  ( ( A  X.  A )  Er  A  <->  ( Rel  ( A  X.  A
)  /\  dom  ( A  X.  A )  =  A  /\  ( `' ( A  X.  A
)  u.  ( ( A  X.  A )  o.  ( A  X.  A ) ) ) 
C_  ( A  X.  A ) ) )
161, 2, 14, 15mpbir3an 1148 1  |-  ( A  X.  A )  Er  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    u. cun 3039    C_ wss 3041    X. cxp 4507   `'ccnv 4508   dom cdm 4509    o. ccom 4513   Rel wrel 4514    Er wer 6394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-er 6397
This theorem is referenced by: (None)
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