MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmpti Unicode version

Theorem fvmpti 5744
Description: Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
Hypotheses
Ref Expression
fvmptg.1  |-  ( x  =  A  ->  B  =  C )
fvmptg.2  |-  F  =  ( x  e.  D  |->  B )
Assertion
Ref Expression
fvmpti  |-  ( A  e.  D  ->  ( F `  A )  =  (  _I  `  C
) )
Distinct variable groups:    x, A    x, C    x, D
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fvmpti
StepHypRef Expression
1 fvmptg.1 . . . 4  |-  ( x  =  A  ->  B  =  C )
2 fvmptg.2 . . . 4  |-  F  =  ( x  e.  D  |->  B )
31, 2fvmptg 5743 . . 3  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  ( F `  A
)  =  C )
4 fvi 5722 . . . 4  |-  ( C  e.  _V  ->  (  _I  `  C )  =  C )
54adantl 453 . . 3  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  (  _I  `  C
)  =  C )
63, 5eqtr4d 2422 . 2  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  ( F `  A
)  =  (  _I 
`  C ) )
71eleq1d 2453 . . . . . . . 8  |-  ( x  =  A  ->  ( B  e.  _V  <->  C  e.  _V ) )
82dmmpt 5305 . . . . . . . 8  |-  dom  F  =  { x  e.  D  |  B  e.  _V }
97, 8elrab2 3037 . . . . . . 7  |-  ( A  e.  dom  F  <->  ( A  e.  D  /\  C  e. 
_V ) )
109baib 872 . . . . . 6  |-  ( A  e.  D  ->  ( A  e.  dom  F  <->  C  e.  _V ) )
1110notbid 286 . . . . 5  |-  ( A  e.  D  ->  ( -.  A  e.  dom  F  <->  -.  C  e.  _V ) )
12 ndmfv 5695 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
1311, 12syl6bir 221 . . . 4  |-  ( A  e.  D  ->  ( -.  C  e.  _V  ->  ( F `  A
)  =  (/) ) )
1413imp 419 . . 3  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (/) )
15 fvprc 5662 . . . 4  |-  ( -.  C  e.  _V  ->  (  _I  `  C )  =  (/) )
1615adantl 453 . . 3  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  (  _I  `  C )  =  (/) )
1714, 16eqtr4d 2422 . 2  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (  _I  `  C ) )
186, 17pm2.61dan 767 1  |-  ( A  e.  D  ->  ( F `  A )  =  (  _I  `  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899   (/)c0 3571    e. cmpt 4207    _I cid 4434   dom cdm 4818   ` cfv 5394
This theorem is referenced by:  fvmpt2i  5750  fvmptex  5754  sumeq2ii  12414  summolem3  12435  fsumf1o  12444  isumshft  12546  prodeq2ii  25018  prodmolem3  25038  fprodf1o  25051
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fv 5402
  Copyright terms: Public domain W3C validator