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Theorem fvmpti 5601
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 5600 . . 3  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  ( F `  A
)  =  C )
4 fvi 5579 . . . 4  |-  ( C  e.  _V  ->  (  _I  `  C )  =  C )
54adantl 452 . . 3  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  (  _I  `  C
)  =  C )
63, 5eqtr4d 2318 . 2  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  ( F `  A
)  =  (  _I 
`  C ) )
71eleq1d 2349 . . . . . . . 8  |-  ( x  =  A  ->  ( B  e.  _V  <->  C  e.  _V ) )
82dmmpt 5168 . . . . . . . 8  |-  dom  F  =  { x  e.  D  |  B  e.  _V }
97, 8elrab2 2925 . . . . . . 7  |-  ( A  e.  dom  F  <->  ( A  e.  D  /\  C  e. 
_V ) )
109baib 871 . . . . . 6  |-  ( A  e.  D  ->  ( A  e.  dom  F  <->  C  e.  _V ) )
1110notbid 285 . . . . 5  |-  ( A  e.  D  ->  ( -.  A  e.  dom  F  <->  -.  C  e.  _V ) )
12 ndmfv 5552 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
1311, 12syl6bir 220 . . . 4  |-  ( A  e.  D  ->  ( -.  C  e.  _V  ->  ( F `  A
)  =  (/) ) )
1413imp 418 . . 3  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (/) )
15 fvprc 5519 . . . 4  |-  ( -.  C  e.  _V  ->  (  _I  `  C )  =  (/) )
1615adantl 452 . . 3  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  (  _I  `  C )  =  (/) )
1714, 16eqtr4d 2318 . 2  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (  _I  `  C ) )
186, 17pm2.61dan 766 1  |-  ( A  e.  D  ->  ( F `  A )  =  (  _I  `  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455    e. cmpt 4077    _I cid 4304   dom cdm 4689   ` cfv 5255
This theorem is referenced by:  fvmpt2i  5607  fvmptex  5610  sumeq2ii  12166  summolem3  12187  fsumf1o  12196  isumshft  12298  fvmpt2f  23224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  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-iota 5219  df-fun 5257  df-fv 5263
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