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Theorem ex-opab 23574
Description: Example for df-opab 4348. 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 10392 . . 3  |-  3  e.  CC
2 4cn 10395 . . 3  |-  4  e.  CC
3 3p1e4 10443 . . 3  |-  ( 3  +  1 )  =  4
41elexi 2980 . . . 4  |-  3  e.  _V
52elexi 2980 . . . 4  |-  4  e.  _V
6 eleq1 2501 . . . . 5  |-  ( x  =  3  ->  (
x  e.  CC  <->  3  e.  CC ) )
7 oveq1 6097 . . . . . 6  |-  ( x  =  3  ->  (
x  +  1 )  =  ( 3  +  1 ) )
87eqeq1d 2449 . . . . 5  |-  ( x  =  3  ->  (
( x  +  1 )  =  y  <->  ( 3  +  1 )  =  y ) )
96, 83anbi13d 1286 . . . 4  |-  ( x  =  3  ->  (
( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1
)  =  y )  <-> 
( 3  e.  CC  /\  y  e.  CC  /\  ( 3  +  1 )  =  y ) ) )
10 eleq1 2501 . . . . 5  |-  ( y  =  4  ->  (
y  e.  CC  <->  4  e.  CC ) )
11 eqeq2 2450 . . . . 5  |-  ( y  =  4  ->  (
( 3  +  1 )  =  y  <->  ( 3  +  1 )  =  4 ) )
1210, 113anbi23d 1287 . . . 4  |-  ( y  =  4  ->  (
( 3  e.  CC  /\  y  e.  CC  /\  ( 3  +  1 )  =  y )  <-> 
( 3  e.  CC  /\  4  e.  CC  /\  ( 3  +  1 )  =  4 ) ) )
13 eqid 2441 . . . 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 4609 . . 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 1165 . 2  |-  3 { <. x ,  y
>.  |  ( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1 )  =  y ) } 4
16 breq 4291 . 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 233 1  |-  ( R  =  { <. x ,  y >.  |  ( x  e.  CC  /\  y  e.  CC  /\  (
x  +  1 )  =  y ) }  ->  3 R 4 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 960    = wceq 1364    e. wcel 1761   class class class wbr 4289   {copab 4346  (class class class)co 6090   CCcc 9276   1c1 9279    + caddc 9281   3c3 10368   4c4 10369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-i2m1 9346  ax-1ne0 9347  ax-rrecex 9350  ax-cnre 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-iota 5378  df-fv 5423  df-ov 6093  df-2 10376  df-3 10377  df-4 10378
This theorem is referenced by: (None)
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