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Theorem ovmpt3rabdm 6508
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 3335 . . . . . 6  |-  ( y  =  Y  ->  ( ph 
<-> 
[. Y  /  y ]. ph ) )
5 sbceq1a 3335 . . . . . 6  |-  ( x  =  X  ->  ( [. Y  /  y ]. ph  <->  [. X  /  x ]. [. Y  /  y ]. ph ) )
64, 5sylan9bbr 698 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  [. X  /  x ]. [. Y  / 
y ]. ph ) )
7 nfsbc1v 3344 . . . . 5  |-  F/ x [. X  /  x ]. [. Y  /  y ]. ph
8 nfcv 2616 . . . . . 6  |-  F/_ y X
9 nfsbc1v 3344 . . . . . 6  |-  F/ y
[. Y  /  y ]. ph
108, 9nfsbc 3346 . . . . 5  |-  F/ y
[. X  /  x ]. [. Y  /  y ]. ph
111, 2, 3, 6, 7, 10ovmpt3rab1 6507 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U )  ->  ( X O Y )  =  ( z  e.  K  |->  { a  e.  L  |  [. X  /  x ]. [. Y  /  y ]. ph }
) )
1211adantr 463 . . 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 } ) )
1312dmeqd 5194 . 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 }
) )
14 rabexg 4587 . . . . 5  |-  ( L  e.  T  ->  { a  e.  L  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
1514adantl 464 . . . 4  |-  ( ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U
)  /\  L  e.  T )  ->  { a  e.  L  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
1615ralrimivw 2869 . . 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 )
17 dmmptg 5487 . . 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 )
1816, 17syl 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 )
1913, 18eqtrd 2495 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   _Vcvv 3106   [.wsbc 3324    |-> cmpt 4497   dom cdm 4988  (class class class)co 6270    |-> cmpt2 6272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275
This theorem is referenced by:  elovmpt3rab1  6509
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