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Theorem ex-opab 20835
Description: Example for df-opab 4094. 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 9834 . . 3  |-  3  e.  CC
2 4cn 9836 . . 3  |-  4  e.  CC
3 3p1e4 9864 . . 3  |-  ( 3  +  1 )  =  4
41elexi 2810 . . . 4  |-  3  e.  _V
52elexi 2810 . . . 4  |-  4  e.  _V
6 eleq1 2356 . . . . 5  |-  ( x  =  3  ->  (
x  e.  CC  <->  3  e.  CC ) )
7 oveq1 5881 . . . . . 6  |-  ( x  =  3  ->  (
x  +  1 )  =  ( 3  +  1 ) )
87eqeq1d 2304 . . . . 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 2356 . . . . 5  |-  ( y  =  4  ->  (
y  e.  CC  <->  4  e.  CC ) )
11 eqeq2 2305 . . . . 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 2296 . . . 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 4303 . . 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 4041 . 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 1632    e. wcel 1696   class class class wbr 4039   {copab 4092  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756   3c3 9812   4c4 9813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-iota 5235  df-fv 5279  df-ov 5877  df-2 9820  df-3 9821  df-4 9822
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