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Theorem xpiderm 6164
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 4434 . . 3  |-  Rel  ( A  X.  A )
21a1i 9 . 2  |-  ( E. x  x  e.  A  ->  Rel  ( A  X.  A ) )
3 dmxpm 4542 . 2  |-  ( E. x  x  e.  A  ->  dom  ( A  X.  A )  =  A )
4 cnvxp 4729 . . . 4  |-  `' ( A  X.  A )  =  ( A  X.  A )
5 xpidtr 4702 . . . 4  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  C_  ( A  X.  A
)
6 uneq1 3087 . . . . 5  |-  ( `' ( A  X.  A
)  =  ( A  X.  A )  -> 
( `' ( A  X.  A )  u.  ( A  X.  A
) )  =  ( ( A  X.  A
)  u.  ( A  X.  A ) ) )
7 unss2 3111 . . . . 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 3083 . . . . . 6  |-  ( ( A  X.  A )  u.  ( A  X.  A ) )  =  ( A  X.  A
)
9 eqtr 2057 . . . . . . 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 2964 . . . . . . . 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 132 . . . . . . 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 401 . . . . 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 34 . . . 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 6093 . 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 1088 1  |-  ( E. x  x  e.  A  ->  ( A  X.  A
)  Er  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393    u. cun 2912    C_ wss 2914    X. cxp 4330   `'ccnv 4331   dom cdm 4332    o. ccom 4336   Rel wrel 4337    Er wer 6090
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-br 3762  df-opab 3816  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-er 6093
This theorem is referenced by: (None)
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