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Theorem dfrel2 5080
Description: Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
dfrel2  |-  ( Rel 
R  <->  `' `' R  =  R
)

Proof of Theorem dfrel2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5007 . . 3  |-  Rel  `' `' R
2 vex 2741 . . . . . 6  |-  x  e. 
_V
3 vex 2741 . . . . . 6  |-  y  e. 
_V
42, 3opelcnv 4810 . . . . 5  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. y ,  x >.  e.  `' R )
53, 2opelcnv 4810 . . . . 5  |-  ( <.
y ,  x >.  e.  `' R  <->  <. x ,  y
>.  e.  R )
64, 5bitri 184 . . . 4  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. x ,  y
>.  e.  R )
76eqrelriv 4720 . . 3  |-  ( ( Rel  `' `' R  /\  Rel  R )  ->  `' `' R  =  R
)
81, 7mpan 424 . 2  |-  ( Rel 
R  ->  `' `' R  =  R )
9 releq 4709 . . 3  |-  ( `' `' R  =  R  ->  ( Rel  `' `' R 
<->  Rel  R ) )
101, 9mpbii 148 . 2  |-  ( `' `' R  =  R  ->  Rel  R )
118, 10impbii 126 1  |-  ( Rel 
R  <->  `' `' R  =  R
)
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1353    e. wcel 2148   <.cop 3596   `'ccnv 4626   Rel wrel 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635
This theorem is referenced by:  dfrel4v  5081  cnvcnv  5082  cnveqb  5085  dfrel3  5087  cnvcnvres  5093  cnvsn  5112  cores2  5142  co01  5144  coi2  5146  relcnvtr  5149  relcnvexb  5169  funcnvres2  5292  f1cnvcnv  5433  f1ocnv  5475  f1ocnvb  5476  f1ococnv1  5491  isores1  5815  cnvf1o  6226  tposf12  6270  ssenen  6851  relcnvfi  6940  caseinl  7090  caseinr  7091  fsumcnv  11445  fprodcnv  11633  structcnvcnv  12478  hmeocnv  13810  hmeocnvb  13821
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