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

Theorem fvmpti 5563
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 5562 . . 3  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  ( F `  A
)  =  C )
4 fvi 5541 . . . 4  |-  ( C  e.  _V  ->  (  _I  `  C )  =  C )
54adantl 454 . . 3  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  (  _I  `  C
)  =  C )
63, 5eqtr4d 2320 . 2  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  ( F `  A
)  =  (  _I 
`  C ) )
71eleq1d 2351 . . . . . . . 8  |-  ( x  =  A  ->  ( B  e.  _V  <->  C  e.  _V ) )
82dmmpt 5167 . . . . . . . 8  |-  dom  F  =  { x  e.  D  |  B  e.  _V }
97, 8elrab2 2927 . . . . . . 7  |-  ( A  e.  dom  F  <->  ( A  e.  D  /\  C  e. 
_V ) )
109baib 873 . . . . . 6  |-  ( A  e.  D  ->  ( A  e.  dom  F  <->  C  e.  _V ) )
1110notbid 287 . . . . 5  |-  ( A  e.  D  ->  ( -.  A  e.  dom  F  <->  -.  C  e.  _V ) )
12 ndmfv 5514 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
1311, 12syl6bir 222 . . . 4  |-  ( A  e.  D  ->  ( -.  C  e.  _V  ->  ( F `  A
)  =  (/) ) )
1413imp 420 . . 3  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (/) )
15 fvprc 5483 . . . 4  |-  ( -.  C  e.  _V  ->  (  _I  `  C )  =  (/) )
1615adantl 454 . . 3  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  (  _I  `  C )  =  (/) )
1714, 16eqtr4d 2320 . 2  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (  _I  `  C ) )
186, 17pm2.61dan 768 1  |-  ( A  e.  D  ->  ( F `  A )  =  (  _I  `  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   _Vcvv 2790   (/)c0 3457    e. cmpt 4079    _I cid 4304   dom cdm 4689   ` cfv 5222
This theorem is referenced by:  fvmpt2i  5569  fvmptex  5572  sumeq2ii  12161  summolem3  12182  fsumf1o  12191  isumshft  12293
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fv 5230
  Copyright terms: Public domain W3C validator