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Theorem dfrel4 23987
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5731 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.)
Hypotheses
Ref Expression
dfrel4.1  |-  F/_ x R
dfrel4.2  |-  F/_ y R
Assertion
Ref Expression
dfrel4  |-  ( Rel 
R  <->  R  =  { <. x ,  y >.  |  x R y } )
Distinct variable group:    x, y
Allowed substitution hints:    R( x, y)

Proof of Theorem dfrel4
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrel4v 5281 . 2  |-  ( Rel 
R  <->  R  =  { <. a ,  b >.  |  a R b } )
2 nfcv 2540 . . . . 5  |-  F/_ x
a
3 dfrel4.1 . . . . 5  |-  F/_ x R
4 nfcv 2540 . . . . 5  |-  F/_ x
b
52, 3, 4nfbr 4216 . . . 4  |-  F/ x  a R b
6 nfcv 2540 . . . . 5  |-  F/_ y
a
7 dfrel4.2 . . . . 5  |-  F/_ y R
8 nfcv 2540 . . . . 5  |-  F/_ y
b
96, 7, 8nfbr 4216 . . . 4  |-  F/ y  a R b
10 nfv 1626 . . . 4  |-  F/ a  x R y
11 nfv 1626 . . . 4  |-  F/ b  x R y
12 breq12 4177 . . . 4  |-  ( ( a  =  x  /\  b  =  y )  ->  ( a R b  <-> 
x R y ) )
135, 9, 10, 11, 12cbvopab 4236 . . 3  |-  { <. a ,  b >.  |  a R b }  =  { <. x ,  y
>.  |  x R
y }
1413eqeq2i 2414 . 2  |-  ( R  =  { <. a ,  b >.  |  a R b }  <->  R  =  { <. x ,  y
>.  |  x R
y } )
151, 14bitri 241 1  |-  ( Rel 
R  <->  R  =  { <. x ,  y >.  |  x R y } )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649   F/_wnfc 2527   class class class wbr 4172   {copab 4225   Rel wrel 4842
This theorem is referenced by:  feqmptdf  24028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845
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