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Theorem funbrfv 5426
Description: The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
funbrfv  |-  ( Fun 
F  ->  ( A F B  ->  ( F `
 A )  =  B ) )

Proof of Theorem funbrfv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 funrel 5108 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 brrelex2 4548 . . . 4  |-  ( ( Rel  F  /\  A F B )  ->  B  e.  _V )
31, 2sylan 279 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  B  e.  _V )
4 breq2 3901 . . . . . 6  |-  ( y  =  B  ->  ( A F y  <->  A F B ) )
54anbi2d 457 . . . . 5  |-  ( y  =  B  ->  (
( Fun  F  /\  A F y )  <->  ( Fun  F  /\  A F B ) ) )
6 eqeq2 2125 . . . . 5  |-  ( y  =  B  ->  (
( F `  A
)  =  y  <->  ( F `  A )  =  B ) )
75, 6imbi12d 233 . . . 4  |-  ( y  =  B  ->  (
( ( Fun  F  /\  A F y )  ->  ( F `  A )  =  y )  <->  ( ( Fun 
F  /\  A F B )  ->  ( F `  A )  =  B ) ) )
8 funeu 5116 . . . . . 6  |-  ( ( Fun  F  /\  A F y )  ->  E! y  A F
y )
9 tz6.12-1 5414 . . . . . 6  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
108, 9sylan2 282 . . . . 5  |-  ( ( A F y  /\  ( Fun  F  /\  A F y ) )  ->  ( F `  A )  =  y )
1110anabss7 555 . . . 4  |-  ( ( Fun  F  /\  A F y )  -> 
( F `  A
)  =  y )
127, 11vtoclg 2718 . . 3  |-  ( B  e.  _V  ->  (
( Fun  F  /\  A F B )  -> 
( F `  A
)  =  B ) )
133, 12mpcom 36 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( F `  A )  =  B )
1413ex 114 1  |-  ( Fun 
F  ->  ( A F B  ->  ( F `
 A )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   E!weu 1975   _Vcvv 2658   class class class wbr 3897   Rel wrel 4512   Fun wfun 5085   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099
This theorem is referenced by:  funopfv  5427  fnbrfvb  5428  fvelima  5439  fvi  5444  fmptco  5552  fliftfun  5663  fliftval  5667  tfrlem5  6177  sum0  11108  isumz  11109  fsumsersdc  11115  isumclim  11141  dvaddxx  12742  dvmulxx  12743  dvcj  12748  dvrecap  12752  dvef  12762
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