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Theorem ex-opab 20819
Description: Example for df-opab 4078. 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 9818 . . 3  |-  3  e.  CC
2 4cn 9820 . . 3  |-  4  e.  CC
3 3p1e4 9848 . . 3  |-  ( 3  +  1 )  =  4
41elexi 2797 . . . 4  |-  3  e.  _V
52elexi 2797 . . . 4  |-  4  e.  _V
6 eleq1 2343 . . . . 5  |-  ( x  =  3  ->  (
x  e.  CC  <->  3  e.  CC ) )
7 oveq1 5865 . . . . . 6  |-  ( x  =  3  ->  (
x  +  1 )  =  ( 3  +  1 ) )
87eqeq1d 2291 . . . . 5  |-  ( x  =  3  ->  (
( x  +  1 )  =  y  <->  ( 3  +  1 )  =  y ) )
96, 83anbi13d 1254 . . . 4  |-  ( x  =  3  ->  (
( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1
)  =  y )  <-> 
( 3  e.  CC  /\  y  e.  CC  /\  ( 3  +  1 )  =  y ) ) )
10 eleq1 2343 . . . . 5  |-  ( y  =  4  ->  (
y  e.  CC  <->  4  e.  CC ) )
11 eqeq2 2292 . . . . 5  |-  ( y  =  4  ->  (
( 3  +  1 )  =  y  <->  ( 3  +  1 )  =  4 ) )
1210, 113anbi23d 1255 . . . 4  |-  ( y  =  4  ->  (
( 3  e.  CC  /\  y  e.  CC  /\  ( 3  +  1 )  =  y )  <-> 
( 3  e.  CC  /\  4  e.  CC  /\  ( 3  +  1 )  =  4 ) ) )
13 eqid 2283 . . . 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 4287 . . 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 1134 . 2  |-  3 { <. x ,  y
>.  |  ( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1 )  =  y ) } 4
16 breq 4025 . 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 224 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 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   {copab 4076  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740   3c3 9796   4c4 9797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263  df-ov 5861  df-2 9804  df-3 9805  df-4 9806
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