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Theorem elfvmptrab1 5970
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 ne0i 3791 . . 3  |-  ( Y  e.  ( F `  X )  ->  ( F `  X )  =/=  (/) )
2 ndmfv 5890 . . . 4  |-  ( -.  X  e.  dom  F  ->  ( F `  X
)  =  (/) )
32necon1ai 2698 . . 3  |-  ( ( F `  X )  =/=  (/)  ->  X  e.  dom  F )
4 elfvmptrab1.f . . . . . . . 8  |-  F  =  ( x  e.  V  |->  { y  e.  [_ x  /  m ]_ M  |  ph } )
54dmmptss 5503 . . . . . . 7  |-  dom  F  C_  V
65sseli 3500 . . . . . 6  |-  ( X  e.  dom  F  ->  X  e.  V )
7 elfvmptrab1.v . . . . . . 7  |-  ( X  e.  V  ->  [_ X  /  m ]_ M  e. 
_V )
8 rabexg 4597 . . . . . . 7  |-  ( [_ X  /  m ]_ M  e.  _V  ->  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }  e.  _V )
96, 7, 83syl 20 . . . . . 6  |-  ( X  e.  dom  F  ->  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }  e.  _V )
10 nfcv 2629 . . . . . . 7  |-  F/_ x X
11 nfsbc1v 3351 . . . . . . . 8  |-  F/ x [. X  /  x ]. ph
12 nfcv 2629 . . . . . . . . 9  |-  F/_ x M
1310, 12nfcsb 3453 . . . . . . . 8  |-  F/_ x [_ X  /  m ]_ M
1411, 13nfrab 3043 . . . . . . 7  |-  F/_ x { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
15 csbeq1 3438 . . . . . . . 8  |-  ( x  =  X  ->  [_ x  /  m ]_ M  = 
[_ X  /  m ]_ M )
16 sbceq1a 3342 . . . . . . . 8  |-  ( x  =  X  ->  ( ph 
<-> 
[. X  /  x ]. ph ) )
1715, 16rabeqbidv 3108 . . . . . . 7  |-  ( x  =  X  ->  { y  e.  [_ x  /  m ]_ M  |  ph }  =  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
)
1810, 14, 17, 4fvmptf 5966 . . . . . 6  |-  ( ( X  e.  V  /\  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }  e.  _V )  ->  ( F `  X )  =  {
y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph } )
196, 9, 18syl2anc 661 . . . . 5  |-  ( X  e.  dom  F  -> 
( F `  X
)  =  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
)
2019eleq2d 2537 . . . 4  |-  ( X  e.  dom  F  -> 
( Y  e.  ( F `  X )  <-> 
Y  e.  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
) )
21 elrabi 3258 . . . . . 6  |-  ( Y  e.  { y  e. 
[_ X  /  m ]_ M  |  [. X  /  x ]. ph }  ->  Y  e.  [_ X  /  m ]_ M )
226, 21anim12i 566 . . . . 5  |-  ( ( X  e.  dom  F  /\  Y  e.  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
)  ->  ( X  e.  V  /\  Y  e. 
[_ X  /  m ]_ M ) )
2322ex 434 . . . 4  |-  ( X  e.  dom  F  -> 
( Y  e.  {
y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }  ->  ( X  e.  V  /\  Y  e.  [_ X  /  m ]_ M ) ) )
2420, 23sylbid 215 . . 3  |-  ( X  e.  dom  F  -> 
( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e. 
[_ X  /  m ]_ M ) ) )
251, 3, 243syl 20 . 2  |-  ( Y  e.  ( F `  X )  ->  ( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e.  [_ X  /  m ]_ M ) ) )
2625pm2.43i 47 1  |-  ( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e.  [_ X  /  m ]_ M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818   _Vcvv 3113   [.wsbc 3331   [_csb 3435   (/)c0 3785    |-> cmpt 4505   dom cdm 4999   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596
This theorem is referenced by:  elfvmptrab  5971
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