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Theorem ovmpt3rabdm 6517
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 3342 . . . . . 6  |-  ( y  =  Y  ->  ( ph 
<-> 
[. Y  /  y ]. ph ) )
5 sbceq1a 3342 . . . . . 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 2629 . . . . . 6  |-  F/_ x X
87nfsbc1 3350 . . . . 5  |-  F/ x [. X  /  x ]. [. Y  /  y ]. ph
9 nfcv 2629 . . . . . 6  |-  F/_ y X
10 nfcv 2629 . . . . . . 7  |-  F/_ y Y
1110nfsbc1 3350 . . . . . 6  |-  F/ y
[. Y  /  y ]. ph
129, 11nfsbc 3353 . . . . 5  |-  F/ y
[. X  /  x ]. [. Y  /  y ]. ph
131, 2, 3, 6, 8, 12ovmpt3rab1 6516 . . . 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 5203 . 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 4597 . . . . 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 2879 . . 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 5502 . . 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 2508 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113   [.wsbc 3331    |-> cmpt 4505   dom cdm 4999  (class class class)co 6282    |-> cmpt2 6284
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  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-reu 2821  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-iun 4327  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287
This theorem is referenced by:  elovmpt3rab1  6518
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