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Theorem elfvmptrab 5519
Description: Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Hypotheses
Ref Expression
elfvmptrab.f  |-  F  =  ( x  e.  V  |->  { y  e.  M  |  ph } )
elfvmptrab.v  |-  ( X  e.  V  ->  M  e.  _V )
Assertion
Ref Expression
elfvmptrab  |-  ( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e.  M )
)
Distinct variable groups:    x, M, y   
x, V    x, X, y    y, Y
Allowed substitution hints:    ph( x, y)    F( x, y)    V( y)    Y( x)

Proof of Theorem elfvmptrab
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 elfvmptrab.f . . . 4  |-  F  =  ( x  e.  V  |->  { y  e.  M  |  ph } )
2 csbconstg 3016 . . . . . . 7  |-  ( x  e.  V  ->  [_ x  /  m ]_ M  =  M )
32eqcomd 2145 . . . . . 6  |-  ( x  e.  V  ->  M  =  [_ x  /  m ]_ M )
4 rabeq 2678 . . . . . 6  |-  ( M  =  [_ x  /  m ]_ M  ->  { y  e.  M  |  ph }  =  { y  e.  [_ x  /  m ]_ M  |  ph }
)
53, 4syl 14 . . . . 5  |-  ( x  e.  V  ->  { y  e.  M  |  ph }  =  { y  e.  [_ x  /  m ]_ M  |  ph }
)
65mpteq2ia 4017 . . . 4  |-  ( x  e.  V  |->  { y  e.  M  |  ph } )  =  ( x  e.  V  |->  { y  e.  [_ x  /  m ]_ M  |  ph } )
71, 6eqtri 2160 . . 3  |-  F  =  ( x  e.  V  |->  { y  e.  [_ x  /  m ]_ M  |  ph } )
8 csbconstg 3016 . . . 4  |-  ( X  e.  V  ->  [_ X  /  m ]_ M  =  M )
9 elfvmptrab.v . . . 4  |-  ( X  e.  V  ->  M  e.  _V )
108, 9eqeltrd 2216 . . 3  |-  ( X  e.  V  ->  [_ X  /  m ]_ M  e. 
_V )
117, 10elfvmptrab1 5518 . 2  |-  ( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e.  [_ X  /  m ]_ M ) )
128eleq2d 2209 . . . 4  |-  ( X  e.  V  ->  ( Y  e.  [_ X  /  m ]_ M  <->  Y  e.  M ) )
1312biimpd 143 . . 3  |-  ( X  e.  V  ->  ( Y  e.  [_ X  /  m ]_ M  ->  Y  e.  M ) )
1413imdistani 441 . 2  |-  ( ( X  e.  V  /\  Y  e.  [_ X  /  m ]_ M )  -> 
( X  e.  V  /\  Y  e.  M
) )
1511, 14syl 14 1  |-  ( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e.  M )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   {crab 2420   _Vcvv 2686   [_csb 3003    |-> cmpt 3992   ` cfv 5126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-mpt 3994  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fv 5134
This theorem is referenced by: (None)
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