Theorem mpt2ndm0 6500

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.

Step	Hyp	Ref	Expression
1		mpt2ndm0.f	. . . . 5 |- F = ( x e. X , y e. Y |-> C )
2		df-mpt2 6289	. . . . 5 |- ( x e. X , y e. Y |-> C ) = { <. <. x , y >. , z >. | ( ( x e. X /\ y e. Y ) /\ z = C ) }
3	1, 2	eqtri 2496	. . . 4 |- F = { <. <. x , y >. , z >. | ( ( x e. X /\ y e. Y ) /\ z = C ) }
4	3	dmeqi 5204	. . 3 |- dom F = dom { <. <. x , y >. , z >. | ( ( x e. X /\ y e. Y ) /\ z = C ) }
5		dmoprabss 6368	. . 3 |- dom { <. <. x , y >. , z >. | ( ( x e. X /\ y e. Y ) /\ z = C ) } C_ ( X X. Y )
6	4, 5	eqsstri 3534	. 2 |- dom F C_ ( X X. Y )
7		nssdmovg 6441	. 2 |- ( ( dom F C_ ( X X. Y ) /\ -. ( V e. X /\ W e. Y ) ) -> ( V F W ) = (/) )
8	6, 7	mpan 670	1 |- ( -. ( V e. X /\ W e. Y ) -> ( V F W ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   C_ wss 3476   (/)c0 3785   X. cxp 4997   dom cdm 4999  (class class class)co 6284   {coprab 6285   |-> cmpt2 6286

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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  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-rab 2823  df-v 3115  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-br 4448  df-opab 4506  df-xp 5005  df-dm 5009  df-iota 5551  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289

This theorem is referenced by:  elovmpt3imp  6517  brovex  6950  fullfunc  15133  fthfunc  15134  natfval  15173  evlval  17992  matbas0  18707  matrcl  18709  marrepfval  18857  marepvfval  18862  minmar1fval  18943  hmeofval  20022  nghmfval  20992  uvtxisvtx  24194  uvtx0  24195  uvtx01vtx  24196  2wlkonot3v  24579  2spthonot3v  24580