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Theorem fvmpti 5856
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 5855 . . 3  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  ( F `  A
)  =  C )
4 fvi 5831 . . . 4  |-  ( C  e.  _V  ->  (  _I  `  C )  =  C )
54adantl 464 . . 3  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  (  _I  `  C
)  =  C )
63, 5eqtr4d 2426 . 2  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  ( F `  A
)  =  (  _I 
`  C ) )
71eleq1d 2451 . . . . . . . 8  |-  ( x  =  A  ->  ( B  e.  _V  <->  C  e.  _V ) )
82dmmpt 5410 . . . . . . . 8  |-  dom  F  =  { x  e.  D  |  B  e.  _V }
97, 8elrab2 3184 . . . . . . 7  |-  ( A  e.  dom  F  <->  ( A  e.  D  /\  C  e. 
_V ) )
109baib 901 . . . . . 6  |-  ( A  e.  D  ->  ( A  e.  dom  F  <->  C  e.  _V ) )
1110notbid 292 . . . . 5  |-  ( A  e.  D  ->  ( -.  A  e.  dom  F  <->  -.  C  e.  _V ) )
12 ndmfv 5798 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
1311, 12syl6bir 229 . . . 4  |-  ( A  e.  D  ->  ( -.  C  e.  _V  ->  ( F `  A
)  =  (/) ) )
1413imp 427 . . 3  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (/) )
15 fvprc 5768 . . . 4  |-  ( -.  C  e.  _V  ->  (  _I  `  C )  =  (/) )
1615adantl 464 . . 3  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  (  _I  `  C )  =  (/) )
1714, 16eqtr4d 2426 . 2  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (  _I  `  C ) )
186, 17pm2.61dan 789 1  |-  ( A  e.  D  ->  ( F `  A )  =  (  _I  `  C
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   _Vcvv 3034   (/)c0 3711    |-> cmpt 4425    _I cid 4704   dom cdm 4913   ` cfv 5496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fv 5504
This theorem is referenced by:  fvmpt2i  5864  fvmptex  5868  sumeq2ii  13517  summolem3  13538  fsumf1o  13547  isumshft  13653  prodeq2ii  13722  prodmolem3  13742  fprodf1o  13755
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