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Theorem xpiderm 6243
Description: A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by Jim Kingdon, 22-Aug-2019.)
Assertion
Ref Expression
xpiderm  |-  ( E. x  x  e.  A  ->  ( A  X.  A
)  Er  A )
Distinct variable group:    x, A

Proof of Theorem xpiderm
StepHypRef Expression
1 relxp 4475 . . 3  |-  Rel  ( A  X.  A )
21a1i 9 . 2  |-  ( E. x  x  e.  A  ->  Rel  ( A  X.  A ) )
3 dmxpm 4583 . 2  |-  ( E. x  x  e.  A  ->  dom  ( A  X.  A )  =  A )
4 cnvxp 4772 . . . 4  |-  `' ( A  X.  A )  =  ( A  X.  A )
5 xpidtr 4745 . . . 4  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  C_  ( A  X.  A
)
6 uneq1 3120 . . . . 5  |-  ( `' ( A  X.  A
)  =  ( A  X.  A )  -> 
( `' ( A  X.  A )  u.  ( A  X.  A
) )  =  ( ( A  X.  A
)  u.  ( A  X.  A ) ) )
7 unss2 3144 . . . . 5  |-  ( ( ( 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 ) ) )
8 unidm 3116 . . . . . 6  |-  ( ( A  X.  A )  u.  ( A  X.  A ) )  =  ( A  X.  A
)
9 eqtr 2099 . . . . . . 7  |-  ( ( ( `' ( 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 ) )
10 sseq2 3022 . . . . . . . 8  |-  ( ( `' ( 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 ) ) )
1110biimpd 142 . . . . . . 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
) ) )
129, 11syl 14 . . . . . 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
)  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
) ) )
138, 12mpan2 416 . . . . 5  |-  ( ( `' ( 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
) ) )
146, 7, 13syl2im 38 . . . 4  |-  ( `' ( 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 ) ) )
154, 5, 14mp2 16 . . 3  |-  ( `' ( A  X.  A
)  u.  ( ( A  X.  A )  o.  ( A  X.  A ) ) ) 
C_  ( A  X.  A )
1615a1i 9 . 2  |-  ( E. x  x  e.  A  ->  ( `' ( A  X.  A )  u.  ( ( A  X.  A )  o.  ( A  X.  A ) ) )  C_  ( A  X.  A ) )
17 df-er 6172 . 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 ) ) )
182, 3, 16, 17syl3anbrc 1123 1  |-  ( E. x  x  e.  A  ->  ( A  X.  A
)  Er  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434    u. cun 2972    C_ wss 2974    X. cxp 4369   `'ccnv 4370   dom cdm 4371    o. ccom 4375   Rel wrel 4376    Er wer 6169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-er 6172
This theorem is referenced by: (None)
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