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Theorem fvmpti 5500
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 5499 . . 3  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  ( F `  A
)  =  C )
4 fvi 5478 . . . 4  |-  ( C  e.  _V  ->  (  _I  `  C )  =  C )
54adantl 454 . . 3  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  (  _I  `  C
)  =  C )
63, 5eqtr4d 2291 . 2  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  ( F `  A
)  =  (  _I 
`  C ) )
71eleq1d 2322 . . . . . . . 8  |-  ( x  =  A  ->  ( B  e.  _V  <->  C  e.  _V ) )
82dmmpt 5120 . . . . . . . 8  |-  dom  F  =  { x  e.  D  |  B  e.  _V }
97, 8elrab2 2876 . . . . . . 7  |-  ( A  e.  dom  F  <->  ( A  e.  D  /\  C  e. 
_V ) )
109baib 876 . . . . . 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 5451 . . . . 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 5420 . . . 4  |-  ( -.  C  e.  _V  ->  (  _I  `  C )  =  (/) )
1615adantl 454 . . 3  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  (  _I  `  C )  =  (/) )
1714, 16eqtr4d 2291 . 2  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (  _I  `  C ) )
186, 17pm2.61dan 769 1  |-  ( A  e.  D  ->  ( F `  A )  =  (  _I  `  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2740   (/)c0 3397    e. cmpt 4017    _I cid 4241   dom cdm 4626   ` cfv 4638
This theorem is referenced by:  fvmpt2i  5506  fvmptex  5509  sumeq2ii  12096  summolem3  12117  fsumf1o  12126  isumshft  12225
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654
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