Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovmpt3rabdm Structured version   Unicode version

Theorem ovmpt3rabdm 30115
Description: If the value of a function which is the result of an operation defined by the maps-to notation is not empty, the operands and the argument of the function must be sets. (Contributed by AV, 16-May-2019.)
Hypotheses
Ref Expression
ovmpt3rab1.o  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  ( z  e.  M  |->  { a  e.  N  |  ph } ) )
ovmpt3rab1.m  |-  ( ( x  =  X  /\  y  =  Y )  ->  M  =  K )
ovmpt3rab1.n  |-  ( ( x  =  X  /\  y  =  Y )  ->  N  =  L )
Assertion
Ref Expression
ovmpt3rabdm  |-  ( ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U
)  /\  L  e.  T )  ->  dom  ( X O Y )  =  K )
Distinct variable groups:    x, K, y, z    L, a, x, y    N, a    x, V, y    x, W, y   
x, U, y    X, a, x, y, z    Y, a, x, y, z    z, L    z, T    z, U    z, V    z, W
Allowed substitution hints:    ph( x, y, z, a)    T( x, y, a)    U( a)    K( a)    M( x, y, z, a)    N( x, y, z)    O( x, y, z, a)    V( a)    W( a)

Proof of Theorem ovmpt3rabdm
StepHypRef Expression
1 ovmpt3rab1.o . . . . 5  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  ( z  e.  M  |->  { a  e.  N  |  ph } ) )
2 ovmpt3rab1.m . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  M  =  K )
3 ovmpt3rab1.n . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  N  =  L )
4 sbceq1a 3192 . . . . . 6  |-  ( y  =  Y  ->  ( ph 
<-> 
[. Y  /  y ]. ph ) )
5 sbceq1a 3192 . . . . . 6  |-  ( x  =  X  ->  ( [. Y  /  y ]. ph  <->  [. X  /  x ]. [. Y  /  y ]. ph ) )
64, 5sylan9bbr 700 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  [. X  /  x ]. [. Y  / 
y ]. ph ) )
7 nfcv 2574 . . . . . 6  |-  F/_ x X
87nfsbc1 3200 . . . . 5  |-  F/ x [. X  /  x ]. [. Y  /  y ]. ph
9 nfcv 2574 . . . . . 6  |-  F/_ y X
10 nfcv 2574 . . . . . . 7  |-  F/_ y Y
1110nfsbc1 3200 . . . . . 6  |-  F/ y
[. Y  /  y ]. ph
129, 11nfsbc 3203 . . . . 5  |-  F/ y
[. X  /  x ]. [. Y  /  y ]. ph
131, 2, 3, 6, 8, 12ovmpt3rab1 30114 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U )  ->  ( X O Y )  =  ( z  e.  K  |->  { a  e.  L  |  [. X  /  x ]. [. Y  /  y ]. ph }
) )
1413adantr 465 . . 3  |-  ( ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U
)  /\  L  e.  T )  ->  ( X O Y )  =  ( z  e.  K  |->  { a  e.  L  |  [. X  /  x ]. [. Y  /  y ]. ph } ) )
1514dmeqd 5037 . 2  |-  ( ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U
)  /\  L  e.  T )  ->  dom  ( X O Y )  =  dom  ( z  e.  K  |->  { a  e.  L  |  [. X  /  x ]. [. Y  /  y ]. ph }
) )
16 rabexg 4437 . . . . 5  |-  ( L  e.  T  ->  { a  e.  L  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
1716adantl 466 . . . 4  |-  ( ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U
)  /\  L  e.  T )  ->  { a  e.  L  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
1817ralrimivw 2795 . . 3  |-  ( ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U
)  /\  L  e.  T )  ->  A. z  e.  K  { a  e.  L  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
19 dmmptg 5330 . . 3  |-  ( A. z  e.  K  {
a  e.  L  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V  ->  dom  ( z  e.  K  |->  { a  e.  L  |  [. X  /  x ]. [. Y  /  y ]. ph }
)  =  K )
2018, 19syl 16 . 2  |-  ( ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U
)  /\  L  e.  T )  ->  dom  ( z  e.  K  |->  { a  e.  L  |  [. X  /  x ]. [. Y  /  y ]. ph } )  =  K )
2115, 20eqtrd 2470 1  |-  ( ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U
)  /\  L  e.  T )  ->  dom  ( X O Y )  =  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   {crab 2714   _Vcvv 2967   [.wsbc 3181    e. cmpt 4345   dom cdm 4835  (class class class)co 6086    e. cmpt2 6088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091
This theorem is referenced by:  elovmpt3rab1  30116
  Copyright terms: Public domain W3C validator