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Theorem funbrfv 5732
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 5438 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 brrelex2 4884 . . . 4  |-  ( ( Rel  F  /\  A F B )  ->  B  e.  _V )
31, 2sylan 458 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  B  e.  _V )
4 breq2 4184 . . . . . 6  |-  ( y  =  B  ->  ( A F y  <->  A F B ) )
54anbi2d 685 . . . . 5  |-  ( y  =  B  ->  (
( Fun  F  /\  A F y )  <->  ( Fun  F  /\  A F B ) ) )
6 eqeq2 2421 . . . . 5  |-  ( y  =  B  ->  (
( F `  A
)  =  y  <->  ( F `  A )  =  B ) )
75, 6imbi12d 312 . . . 4  |-  ( y  =  B  ->  (
( ( Fun  F  /\  A F y )  ->  ( F `  A )  =  y )  <->  ( ( Fun 
F  /\  A F B )  ->  ( F `  A )  =  B ) ) )
8 funeu 5444 . . . . . 6  |-  ( ( Fun  F  /\  A F y )  ->  E! y  A F
y )
9 tz6.12-1 5714 . . . . . 6  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
108, 9sylan2 461 . . . . 5  |-  ( ( A F y  /\  ( Fun  F  /\  A F y ) )  ->  ( F `  A )  =  y )
1110anabss7 795 . . . 4  |-  ( ( Fun  F  /\  A F y )  -> 
( F `  A
)  =  y )
127, 11vtoclg 2979 . . 3  |-  ( B  e.  _V  ->  (
( Fun  F  /\  A F B )  -> 
( F `  A
)  =  B ) )
133, 12mpcom 34 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( F `  A )  =  B )
1413ex 424 1  |-  ( Fun 
F  ->  ( A F B  ->  ( F `
 A )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   E!weu 2262   _Vcvv 2924   class class class wbr 4180   Rel wrel 4850   Fun wfun 5415   ` cfv 5421
This theorem is referenced by:  funopfv  5733  fnbrfvb  5734  fvelima  5745  fvi  5750  fmptco  5868  fliftfun  6001  fliftval  6005  opabiota  6505  fpwwe2  8482  nqerid  8774  sum0  12478  sumz  12479  fsumsers  12485  isumclim  12504  cnextfvval  18057  dvadd  19787  dvmul  19788  dvco  19794  dvcj  19797  dvrec  19802  dvcnv  19822  dvef  19825  ftc1cn  19888  ulmdv  20280  minvecolem4b  22341  minvecolem4  22343  hlimuni  22702  chscllem4  23103  fmptcof2  24037  ntrivcvgfvn0  25188  ntrivcvgtail  25189  zprodn0  25226  iprodclim  25272  fvtransport  25878  fvray  25987  fvline  25990  ftc1cnnc  26186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429
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