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Theorem erssxp 7326
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 7312 . . 3  |-  ( R  Er  A  ->  Rel  R )
2 relssdmrn 5511 . . 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 7313 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
5 errn 7325 . . 3  |-  ( R  Er  A  ->  ran  R  =  A )
64, 5xpeq12d 5013 . 2  |-  ( R  Er  A  ->  ( dom  R  X.  ran  R
)  =  ( A  X.  A ) )
73, 6sseqtrd 3525 1  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    C_ wss 3461    X. cxp 4986   dom cdm 4988   ran crn 4989   Rel wrel 4993    Er wer 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-er 7303
This theorem is referenced by:  erex  7327  riiner  7376  efgval  16934  qtophaus  28074
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