MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elfvmptrab1 Structured version   Visualization version   Unicode version

Theorem elfvmptrab1 5985
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 3728 . . 3  |-  ( Y  e.  ( F `  X )  ->  ( F `  X )  =/=  (/) )
2 ndmfv 5903 . . . 4  |-  ( -.  X  e.  dom  F  ->  ( F `  X
)  =  (/) )
32necon1ai 2670 . . 3  |-  ( ( F `  X )  =/=  (/)  ->  X  e.  dom  F )
4 elfvmptrab1.f . . . . . . . 8  |-  F  =  ( x  e.  V  |->  { y  e.  [_ x  /  m ]_ M  |  ph } )
54dmmptss 5338 . . . . . . 7  |-  dom  F  C_  V
65sseli 3414 . . . . . 6  |-  ( X  e.  dom  F  ->  X  e.  V )
7 elfvmptrab1.v . . . . . . 7  |-  ( X  e.  V  ->  [_ X  /  m ]_ M  e. 
_V )
8 rabexg 4549 . . . . . . 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 2612 . . . . . . 7  |-  F/_ x X
11 nfsbc1v 3275 . . . . . . . 8  |-  F/ x [. X  /  x ]. ph
12 nfcv 2612 . . . . . . . . 9  |-  F/_ x M
1310, 12nfcsb 3367 . . . . . . . 8  |-  F/_ x [_ X  /  m ]_ M
1411, 13nfrab 2958 . . . . . . 7  |-  F/_ x { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
15 csbeq1 3352 . . . . . . . 8  |-  ( x  =  X  ->  [_ x  /  m ]_ M  = 
[_ X  /  m ]_ M )
16 sbceq1a 3266 . . . . . . . 8  |-  ( x  =  X  ->  ( ph 
<-> 
[. X  /  x ]. ph ) )
1715, 16rabeqbidv 3026 . . . . . . 7  |-  ( x  =  X  ->  { y  e.  [_ x  /  m ]_ M  |  ph }  =  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
)
1810, 14, 17, 4fvmptf 5981 . . . . . 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 673 . . . . 5  |-  ( X  e.  dom  F  -> 
( F `  X
)  =  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
)
2019eleq2d 2534 . . . 4  |-  ( X  e.  dom  F  -> 
( Y  e.  ( F `  X )  <-> 
Y  e.  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
) )
21 elrabi 3181 . . . . . 6  |-  ( Y  e.  { y  e. 
[_ X  /  m ]_ M  |  [. X  /  x ]. ph }  ->  Y  e.  [_ X  /  m ]_ M )
226, 21anim12i 576 . . . . 5  |-  ( ( X  e.  dom  F  /\  Y  e.  { y  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. ph }
)  ->  ( X  e.  V  /\  Y  e. 
[_ X  /  m ]_ M ) )
2322ex 441 . . . 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 223 . . 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 48 1  |-  ( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e.  [_ X  /  m ]_ M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   {crab 2760   _Vcvv 3031   [.wsbc 3255   [_csb 3349   (/)c0 3722    |-> cmpt 4454   dom cdm 4839   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597
This theorem is referenced by:  elfvmptrab  5986
  Copyright terms: Public domain W3C validator