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Theorem elfvmptrab1 5986
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 3773 . . 3  |-  ( Y  e.  ( F `  X )  ->  ( F `  X )  =/=  (/) )
2 ndmfv 5905 . . . 4  |-  ( -.  X  e.  dom  F  ->  ( F `  X
)  =  (/) )
32necon1ai 2662 . . 3  |-  ( ( F `  X )  =/=  (/)  ->  X  e.  dom  F )
4 elfvmptrab1.f . . . . . . . 8  |-  F  =  ( x  e.  V  |->  { y  e.  [_ x  /  m ]_ M  |  ph } )
54dmmptss 5351 . . . . . . 7  |-  dom  F  C_  V
65sseli 3466 . . . . . 6  |-  ( X  e.  dom  F  ->  X  e.  V )
7 elfvmptrab1.v . . . . . . 7  |-  ( X  e.  V  ->  [_ X  /  m ]_ M  e. 
_V )
8 rabexg 4575 . . . . . . 7  |-  ( [_ X  /  m ]_ M  e.  _V  ->  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }  e.  _V )
96, 7, 83syl 18 . . . . . 6  |-  ( X  e.  dom  F  ->  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }  e.  _V )
10 nfcv 2591 . . . . . . 7  |-  F/_ x X
11 nfsbc1v 3325 . . . . . . . 8  |-  F/ x [. X  /  x ]. ph
12 nfcv 2591 . . . . . . . . 9  |-  F/_ x M
1310, 12nfcsb 3419 . . . . . . . 8  |-  F/_ x [_ X  /  m ]_ M
1411, 13nfrab 3017 . . . . . . 7  |-  F/_ x { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
15 csbeq1 3404 . . . . . . . 8  |-  ( x  =  X  ->  [_ x  /  m ]_ M  = 
[_ X  /  m ]_ M )
16 sbceq1a 3316 . . . . . . . 8  |-  ( x  =  X  ->  ( ph 
<-> 
[. X  /  x ]. ph ) )
1715, 16rabeqbidv 3082 . . . . . . 7  |-  ( x  =  X  ->  { y  e.  [_ x  /  m ]_ M  |  ph }  =  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
)
1810, 14, 17, 4fvmptf 5982 . . . . . 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 665 . . . . 5  |-  ( X  e.  dom  F  -> 
( F `  X
)  =  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
)
2019eleq2d 2499 . . . 4  |-  ( X  e.  dom  F  -> 
( Y  e.  ( F `  X )  <-> 
Y  e.  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
) )
21 elrabi 3232 . . . . . 6  |-  ( Y  e.  { y  e. 
[_ X  /  m ]_ M  |  [. X  /  x ]. ph }  ->  Y  e.  [_ X  /  m ]_ M )
226, 21anim12i 568 . . . . 5  |-  ( ( X  e.  dom  F  /\  Y  e.  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
)  ->  ( X  e.  V  /\  Y  e. 
[_ X  /  m ]_ M ) )
2322ex 435 . . . 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 218 . . 3  |-  ( X  e.  dom  F  -> 
( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e. 
[_ X  /  m ]_ M ) ) )
251, 3, 243syl 18 . 2  |-  ( Y  e.  ( F `  X )  ->  ( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e.  [_ X  /  m ]_ M ) ) )
2625pm2.43i 49 1  |-  ( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e.  [_ X  /  m ]_ M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   {crab 2786   _Vcvv 3087   [.wsbc 3305   [_csb 3401   (/)c0 3767    |-> cmpt 4484   dom cdm 4854   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609
This theorem is referenced by:  elfvmptrab  5987
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