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Theorem xpider 7192
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 4968 . 2  |-  Rel  ( A  X.  A )
2 dmxpid 5080 . 2  |-  dom  ( A  X.  A )  =  A
3 cnvxp 5276 . . 3  |-  `' ( A  X.  A )  =  ( A  X.  A )
4 xpidtr 5241 . . 3  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  C_  ( A  X.  A
)
5 uneq1 3524 . . . 4  |-  ( `' ( A  X.  A
)  =  ( A  X.  A )  -> 
( `' ( A  X.  A )  u.  ( A  X.  A
) )  =  ( ( A  X.  A
)  u.  ( A  X.  A ) ) )
6 unss2 3548 . . . 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 3520 . . . . 5  |-  ( ( A  X.  A )  u.  ( A  X.  A ) )  =  ( A  X.  A
)
8 eqtr 2460 . . . . . 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 3399 . . . . . . 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 207 . . . . . 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 16 . . . . 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 671 . . . 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 9 . 2  |-  ( `' ( A  X.  A
)  u.  ( ( A  X.  A )  o.  ( A  X.  A ) ) ) 
C_  ( A  X.  A )
15 df-er 7122 . 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 1170 1  |-  ( A  X.  A )  Er  A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    u. cun 3347    C_ wss 3349    X. cxp 4859   `'ccnv 4860   dom cdm 4861    o. ccom 4865   Rel wrel 4866    Er wer 7119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-br 4314  df-opab 4372  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-er 7122
This theorem is referenced by:  riiner  7194  efglem  16234
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