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Theorem ex-opab 24977
Description: Example for df-opab 4512. 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 10622 . . 3  |-  3  e.  CC
2 4cn 10625 . . 3  |-  4  e.  CC
3 3p1e4 10673 . . 3  |-  ( 3  +  1 )  =  4
41elexi 3128 . . . 4  |-  3  e.  _V
52elexi 3128 . . . 4  |-  4  e.  _V
6 eleq1 2539 . . . . 5  |-  ( x  =  3  ->  (
x  e.  CC  <->  3  e.  CC ) )
7 oveq1 6302 . . . . . 6  |-  ( x  =  3  ->  (
x  +  1 )  =  ( 3  +  1 ) )
87eqeq1d 2469 . . . . 5  |-  ( x  =  3  ->  (
( x  +  1 )  =  y  <->  ( 3  +  1 )  =  y ) )
96, 83anbi13d 1301 . . . 4  |-  ( x  =  3  ->  (
( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1
)  =  y )  <-> 
( 3  e.  CC  /\  y  e.  CC  /\  ( 3  +  1 )  =  y ) ) )
10 eleq1 2539 . . . . 5  |-  ( y  =  4  ->  (
y  e.  CC  <->  4  e.  CC ) )
11 eqeq2 2482 . . . . 5  |-  ( y  =  4  ->  (
( 3  +  1 )  =  y  <->  ( 3  +  1 )  =  4 ) )
1210, 113anbi23d 1302 . . . 4  |-  ( y  =  4  ->  (
( 3  e.  CC  /\  y  e.  CC  /\  ( 3  +  1 )  =  y )  <-> 
( 3  e.  CC  /\  4  e.  CC  /\  ( 3  +  1 )  =  4 ) ) )
13 eqid 2467 . . . 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 4776 . . 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 1178 . 2  |-  3 { <. x ,  y
>.  |  ( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1 )  =  y ) } 4
16 breq 4455 . 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 973    = wceq 1379    e. wcel 1767   class class class wbr 4453   {copab 4510  (class class class)co 6295   CCcc 9502   1c1 9505    + caddc 9507   3c3 10598   4c4 10599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-i2m1 9572  ax-1ne0 9573  ax-rrecex 9576  ax-cnre 9577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-iota 5557  df-fv 5602  df-ov 6298  df-2 10606  df-3 10607  df-4 10608
This theorem is referenced by: (None)
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