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Theorem mpt2ndm0 6415
Description: The value of an operation given by a maps-to rule is the empty set if the arguments are not contained in the base sets of the rule. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Hypothesis
Ref Expression
mpt2ndm0.f  |-  F  =  ( x  e.  X ,  y  e.  Y  |->  C )
Assertion
Ref Expression
mpt2ndm0  |-  ( -.  ( V  e.  X  /\  W  e.  Y
)  ->  ( V F W )  =  (/) )
Distinct variable groups:    x, y, X    x, Y, y
Allowed substitution hints:    C( x, y)    F( x, y)    V( x, y)    W( x, y)

Proof of Theorem mpt2ndm0
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 mpt2ndm0.f . . . . 5  |-  F  =  ( x  e.  X ,  y  e.  Y  |->  C )
2 df-mpt2 6201 . . . . 5  |-  ( x  e.  X ,  y  e.  Y  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  X  /\  y  e.  Y )  /\  z  =  C
) }
31, 2eqtri 2411 . . . 4  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  X  /\  y  e.  Y )  /\  z  =  C ) }
43dmeqi 5117 . . 3  |-  dom  F  =  dom  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  X  /\  y  e.  Y )  /\  z  =  C
) }
5 dmoprabss 6283 . . 3  |-  dom  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  X  /\  y  e.  Y )  /\  z  =  C ) }  C_  ( X  X.  Y
)
64, 5eqsstri 3447 . 2  |-  dom  F  C_  ( X  X.  Y
)
7 nssdmovg 6356 . 2  |-  ( ( dom  F  C_  ( X  X.  Y )  /\  -.  ( V  e.  X  /\  W  e.  Y
) )  ->  ( V F W )  =  (/) )
86, 7mpan 668 1  |-  ( -.  ( V  e.  X  /\  W  e.  Y
)  ->  ( V F W )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826    C_ wss 3389   (/)c0 3711    X. cxp 4911   dom cdm 4913  (class class class)co 6196   {coprab 6197    |-> cmpt2 6198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-xp 4919  df-dm 4923  df-iota 5460  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201
This theorem is referenced by:  elovmpt3imp  6432  bropopvvv  6779  supp0prc  6820  brovex  6868  fullfunc  15312  fthfunc  15313  natfval  15352  evlval  18306  matbas0  18997  matrcl  18999  marrepfval  19147  marepvfval  19152  submafval  19166  minmar1fval  19233  hmeofval  20344  nghmfval  21314  uvtxisvtx  24611  uvtx0  24612  uvtx01vtx  24613  clwwlknprop  24893  2wlkonot3v  24996  2spthonot3v  24997
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