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Theorem xpider 7372
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 5101 . 2  |-  Rel  ( A  X.  A )
2 dmxpid 5213 . 2  |-  dom  ( A  X.  A )  =  A
3 cnvxp 5415 . . 3  |-  `' ( A  X.  A )  =  ( A  X.  A )
4 xpidtr 5380 . . 3  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  C_  ( A  X.  A
)
5 uneq1 3644 . . . 4  |-  ( `' ( A  X.  A
)  =  ( A  X.  A )  -> 
( `' ( A  X.  A )  u.  ( A  X.  A
) )  =  ( ( A  X.  A
)  u.  ( A  X.  A ) ) )
6 unss2 3668 . . . 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 3640 . . . . 5  |-  ( ( A  X.  A )  u.  ( A  X.  A ) )  =  ( A  X.  A
)
8 eqtr 2486 . . . . . 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 3519 . . . . . . 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 7301 . 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 1173 1  |-  ( A  X.  A )  Er  A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    u. cun 3467    C_ wss 3469    X. cxp 4990   `'ccnv 4991   dom cdm 4992    o. ccom 4996   Rel wrel 4997    Er wer 7298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-er 7301
This theorem is referenced by:  riiner  7374  efglem  16523
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