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Theorem ex-opab 21589
Description: Example for df-opab 4209. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
Assertion
Ref Expression
ex-opab  |-  ( R  =  { <. x ,  y >.  |  ( x  e.  CC  /\  y  e.  CC  /\  (
x  +  1 )  =  y ) }  ->  3 R 4 )
Distinct variable group:    x, y
Allowed substitution hints:    R( x, y)

Proof of Theorem ex-opab
StepHypRef Expression
1 3cn 10005 . . 3  |-  3  e.  CC
2 4cn 10007 . . 3  |-  4  e.  CC
3 3p1e4 10037 . . 3  |-  ( 3  +  1 )  =  4
41elexi 2909 . . . 4  |-  3  e.  _V
52elexi 2909 . . . 4  |-  4  e.  _V
6 eleq1 2448 . . . . 5  |-  ( x  =  3  ->  (
x  e.  CC  <->  3  e.  CC ) )
7 oveq1 6028 . . . . . 6  |-  ( x  =  3  ->  (
x  +  1 )  =  ( 3  +  1 ) )
87eqeq1d 2396 . . . . 5  |-  ( x  =  3  ->  (
( x  +  1 )  =  y  <->  ( 3  +  1 )  =  y ) )
96, 83anbi13d 1256 . . . 4  |-  ( x  =  3  ->  (
( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1
)  =  y )  <-> 
( 3  e.  CC  /\  y  e.  CC  /\  ( 3  +  1 )  =  y ) ) )
10 eleq1 2448 . . . . 5  |-  ( y  =  4  ->  (
y  e.  CC  <->  4  e.  CC ) )
11 eqeq2 2397 . . . . 5  |-  ( y  =  4  ->  (
( 3  +  1 )  =  y  <->  ( 3  +  1 )  =  4 ) )
1210, 113anbi23d 1257 . . . 4  |-  ( y  =  4  ->  (
( 3  e.  CC  /\  y  e.  CC  /\  ( 3  +  1 )  =  y )  <-> 
( 3  e.  CC  /\  4  e.  CC  /\  ( 3  +  1 )  =  4 ) ) )
13 eqid 2388 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  CC  /\  y  e.  CC  /\  (
x  +  1 )  =  y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  y  e.  CC  /\  (
x  +  1 )  =  y ) }
144, 5, 9, 12, 13brab 4419 . . 3  |-  ( 3 { <. x ,  y
>.  |  ( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1 )  =  y ) } 4  <-> 
( 3  e.  CC  /\  4  e.  CC  /\  ( 3  +  1 )  =  4 ) )
151, 2, 3, 14mpbir3an 1136 . 2  |-  3 { <. x ,  y
>.  |  ( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1 )  =  y ) } 4
16 breq 4156 . 2  |-  ( R  =  { <. x ,  y >.  |  ( x  e.  CC  /\  y  e.  CC  /\  (
x  +  1 )  =  y ) }  ->  ( 3 R 4  <->  3 { <. x ,  y >.  |  ( x  e.  CC  /\  y  e.  CC  /\  (
x  +  1 )  =  y ) } 4 ) )
1715, 16mpbiri 225 1  |-  ( R  =  { <. x ,  y >.  |  ( x  e.  CC  /\  y  e.  CC  /\  (
x  +  1 )  =  y ) }  ->  3 R 4 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4154   {copab 4207  (class class class)co 6021   CCcc 8922   1c1 8925    + caddc 8927   3c3 9983   4c4 9984
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-i2m1 8992  ax-1ne0 8993  ax-rrecex 8996  ax-cnre 8997
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-iota 5359  df-fv 5403  df-ov 6024  df-2 9991  df-3 9992  df-4 9993
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