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Theorem elfvmptrab1 5610
Description: Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Hypotheses
Ref Expression
elfvmptrab1.f  |-  F  =  ( x  e.  V  |->  { y  e.  [_ x  /  m ]_ M  |  ph } )
elfvmptrab1.v  |-  ( X  e.  V  ->  [_ X  /  m ]_ M  e. 
_V )
Assertion
Ref Expression
elfvmptrab1  |-  ( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e.  [_ X  /  m ]_ M ) )
Distinct variable groups:    x, M, y   
x, V    x, X, y    y, Y    y, m
Allowed substitution hints:    ph( x, y, m)    F( x, y, m)    M( m)    V( y, m)    X( m)    Y( x, m)

Proof of Theorem elfvmptrab1
StepHypRef Expression
1 elfvmptrab1.f . . . . 5  |-  F  =  ( x  e.  V  |->  { y  e.  [_ x  /  m ]_ M  |  ph } )
21funmpt2 5255 . . . 4  |-  Fun  F
3 funrel 5233 . . . 4  |-  ( Fun 
F  ->  Rel  F )
42, 3ax-mp 5 . . 3  |-  Rel  F
5 relelfvdm 5547 . . 3  |-  ( ( Rel  F  /\  Y  e.  ( F `  X
) )  ->  X  e.  dom  F )
64, 5mpan 424 . 2  |-  ( Y  e.  ( F `  X )  ->  X  e.  dom  F )
71dmmptss 5125 . . . . . 6  |-  dom  F  C_  V
87sseli 3151 . . . . 5  |-  ( X  e.  dom  F  ->  X  e.  V )
9 elfvmptrab1.v . . . . . 6  |-  ( X  e.  V  ->  [_ X  /  m ]_ M  e. 
_V )
10 rabexg 4146 . . . . . 6  |-  ( [_ X  /  m ]_ M  e.  _V  ->  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }  e.  _V )
118, 9, 103syl 17 . . . . 5  |-  ( X  e.  dom  F  ->  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }  e.  _V )
12 nfcv 2319 . . . . . 6  |-  F/_ x X
13 nfsbc1v 2981 . . . . . . 7  |-  F/ x [. X  /  x ]. ph
14 nfcv 2319 . . . . . . . 8  |-  F/_ x M
1512, 14nfcsb 3094 . . . . . . 7  |-  F/_ x [_ X  /  m ]_ M
1613, 15nfrabxy 2657 . . . . . 6  |-  F/_ x { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
17 csbeq1 3060 . . . . . . 7  |-  ( x  =  X  ->  [_ x  /  m ]_ M  = 
[_ X  /  m ]_ M )
18 sbceq1a 2972 . . . . . . 7  |-  ( x  =  X  ->  ( ph 
<-> 
[. X  /  x ]. ph ) )
1917, 18rabeqbidv 2732 . . . . . 6  |-  ( x  =  X  ->  { y  e.  [_ x  /  m ]_ M  |  ph }  =  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
)
2012, 16, 19, 1fvmptf 5608 . . . . 5  |-  ( ( X  e.  V  /\  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }  e.  _V )  ->  ( F `  X )  =  {
y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph } )
218, 11, 20syl2anc 411 . . . 4  |-  ( X  e.  dom  F  -> 
( F `  X
)  =  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
)
2221eleq2d 2247 . . 3  |-  ( X  e.  dom  F  -> 
( Y  e.  ( F `  X )  <-> 
Y  e.  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
) )
23 elrabi 2890 . . . . 5  |-  ( Y  e.  { y  e. 
[_ X  /  m ]_ M  |  [. X  /  x ]. ph }  ->  Y  e.  [_ X  /  m ]_ M )
248, 23anim12i 338 . . . 4  |-  ( ( X  e.  dom  F  /\  Y  e.  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
)  ->  ( X  e.  V  /\  Y  e. 
[_ X  /  m ]_ M ) )
2524ex 115 . . 3  |-  ( X  e.  dom  F  -> 
( Y  e.  {
y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }  ->  ( X  e.  V  /\  Y  e.  [_ X  /  m ]_ M ) ) )
2622, 25sylbid 150 . 2  |-  ( X  e.  dom  F  -> 
( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e. 
[_ X  /  m ]_ M ) ) )
276, 26mpcom 36 1  |-  ( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e.  [_ X  /  m ]_ M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   {crab 2459   _Vcvv 2737   [.wsbc 2962   [_csb 3057    |-> cmpt 4064   dom cdm 4626   Rel wrel 4631   Fun wfun 5210   ` cfv 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fv 5224
This theorem is referenced by:  elfvmptrab  5611
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