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Theorem mpt2ndm0 6734
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 6091 . . . . 5  |-  ( x  e.  X ,  y  e.  Y  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  X  /\  y  e.  Y )  /\  z  =  C
) }
31, 2eqtri 2458 . . . 4  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  X  /\  y  e.  Y )  /\  z  =  C ) }
43dmeqi 5036 . . 3  |-  dom  F  =  dom  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  X  /\  y  e.  Y )  /\  z  =  C
) }
5 dmoprabss 6167 . . 3  |-  dom  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  X  /\  y  e.  Y )  /\  z  =  C ) }  C_  ( X  X.  Y
)
64, 5eqsstri 3381 . 2  |-  dom  F  C_  ( X  X.  Y
)
7 nssdmovg 6240 . 2  |-  ( ( dom  F  C_  ( X  X.  Y )  /\  -.  ( V  e.  X  /\  W  e.  Y
) )  ->  ( V F W )  =  (/) )
86, 7mpan 670 1  |-  ( -.  ( V  e.  X  /\  W  e.  Y
)  ->  ( V F W )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3323   (/)c0 3632    X. cxp 4833   dom cdm 4835  (class class class)co 6086   {coprab 6087    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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  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-rab 2719  df-v 2969  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-br 4288  df-opab 4346  df-xp 4841  df-dm 4845  df-iota 5376  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091
This theorem is referenced by:  brovex  6735  fullfunc  14808  fthfunc  14809  natfval  14848  evlval  17585  matbas0  18262  matrcl  18287  marrepfval  18346  marepvfval  18351  minmar1fval  18427  hmeofval  19306  nghmfval  20276  uvtxisvtx  23349  uvtx0  23350  uvtx01vtx  23351  elovmpt3imp  30113  2wlkonot3v  30347  2spthonot3v  30348
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