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Theorem funbrfv 5919
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 5618 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 brrelex2 4894 . . . 4  |-  ( ( Rel  F  /\  A F B )  ->  B  e.  _V )
31, 2sylan 473 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  B  e.  _V )
4 breq2 4430 . . . . . 6  |-  ( y  =  B  ->  ( A F y  <->  A F B ) )
54anbi2d 708 . . . . 5  |-  ( y  =  B  ->  (
( Fun  F  /\  A F y )  <->  ( Fun  F  /\  A F B ) ) )
6 eqeq2 2444 . . . . 5  |-  ( y  =  B  ->  (
( F `  A
)  =  y  <->  ( F `  A )  =  B ) )
75, 6imbi12d 321 . . . 4  |-  ( y  =  B  ->  (
( ( Fun  F  /\  A F y )  ->  ( F `  A )  =  y )  <->  ( ( Fun 
F  /\  A F B )  ->  ( F `  A )  =  B ) ) )
8 funeu 5625 . . . . . 6  |-  ( ( Fun  F  /\  A F y )  ->  E! y  A F
y )
9 tz6.12-1 5897 . . . . . 6  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
108, 9sylan2 476 . . . . 5  |-  ( ( A F y  /\  ( Fun  F  /\  A F y ) )  ->  ( F `  A )  =  y )
1110anabss7 828 . . . 4  |-  ( ( Fun  F  /\  A F y )  -> 
( F `  A
)  =  y )
127, 11vtoclg 3145 . . 3  |-  ( B  e.  _V  ->  (
( Fun  F  /\  A F B )  -> 
( F `  A
)  =  B ) )
133, 12mpcom 37 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( F `  A )  =  B )
1413ex 435 1  |-  ( Fun 
F  ->  ( A F B  ->  ( F `
 A )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   E!weu 2266   _Vcvv 3087   class class class wbr 4426   Rel wrel 4859   Fun wfun 5595   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609
This theorem is referenced by:  funopfv  5920  fnbrfvb  5921  fvelima  5933  fvi  5938  opabiota  5944  fmptco  6071  fliftfun  6220  fliftval  6224  tfrlem5  7106  fpwwe2  9067  nqerid  9357  sum0  13765  sumz  13766  fsumsers  13772  isumclim  13796  ntrivcvgfvn0  13933  ntrivcvgtail  13934  zprodn0  13971  iprodclim  14029  idinv  15645  cnextfvval  21011  cnextfres  21015  dvadd  22771  dvmul  22772  dvco  22778  dvcj  22781  dvrec  22786  dvcnv  22806  dvef  22809  ftc1cn  22872  ulmdv  23223  minvecolem4b  26365  minvecolem4  26367  hlimuni  26726  chscllem4  27128  fmptcof2  28099  fvtransport  30584  fvray  30693  fvline  30696  ftc1cnnc  31720  frege124d  35992
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