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Theorem erssxp 7221
Description: An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
erssxp  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )

Proof of Theorem erssxp
StepHypRef Expression
1 errel 7207 . . 3  |-  ( R  Er  A  ->  Rel  R )
2 relssdmrn 5453 . . 3  |-  ( Rel 
R  ->  R  C_  ( dom  R  X.  ran  R
) )
31, 2syl 16 . 2  |-  ( R  Er  A  ->  R  C_  ( dom  R  X.  ran  R ) )
4 erdm 7208 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
5 errn 7220 . . 3  |-  ( R  Er  A  ->  ran  R  =  A )
64, 5xpeq12d 4960 . 2  |-  ( R  Er  A  ->  ( dom  R  X.  ran  R
)  =  ( A  X.  A ) )
73, 6sseqtrd 3487 1  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    C_ wss 3423    X. cxp 4933   dom cdm 4935   ran crn 4936   Rel wrel 4940    Er wer 7195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-br 4388  df-opab 4446  df-xp 4941  df-rel 4942  df-cnv 4943  df-dm 4945  df-rn 4946  df-er 7198
This theorem is referenced by:  erex  7222  riiner  7270  efgval  16315
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