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Theorem ovmpt3rabdm 30303
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 3298 . . . . . 6  |-  ( y  =  Y  ->  ( ph 
<-> 
[. Y  /  y ]. ph ) )
5 sbceq1a 3298 . . . . . 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 2613 . . . . . 6  |-  F/_ x X
87nfsbc1 3306 . . . . 5  |-  F/ x [. X  /  x ]. [. Y  /  y ]. ph
9 nfcv 2613 . . . . . 6  |-  F/_ y X
10 nfcv 2613 . . . . . . 7  |-  F/_ y Y
1110nfsbc1 3306 . . . . . 6  |-  F/ y
[. Y  /  y ]. ph
129, 11nfsbc 3309 . . . . 5  |-  F/ y
[. X  /  x ]. [. Y  /  y ]. ph
131, 2, 3, 6, 8, 12ovmpt3rab1 30302 . . . 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 5143 . 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 4543 . . . . 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 2826 . . 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 5436 . . 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 2492 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 1370    e. wcel 1758   A.wral 2795   {crab 2799   _Vcvv 3071   [.wsbc 3287    |-> cmpt 4451   dom cdm 4941  (class class class)co 6193    |-> cmpt2 6195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198
This theorem is referenced by:  elovmpt3rab1  30304
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