Theorem mpo0 6074

Description: A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
mpo0	|- ( x e. (/) , y e. B |-> C ) = (/)

Proof of Theorem mpo0

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpo 6006	. 2 |- ( x e. (/) , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. (/) /\ y e. B ) /\ z = C ) }
2		df-oprab 6005	. 2 |- { <. <. x , y >. , z >. | ( ( x e. (/) /\ y e. B ) /\ z = C ) } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) ) }
3		noel 3495	. . . . . . 7 |- -. x e. (/)
4		simprll 537	. . . . . . 7 |- ( ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) ) -> x e. (/) )
5	3, 4	mto 666	. . . . . 6 |- -. ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
6	5	nex 1546	. . . . 5 |- -. E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
7	6	nex 1546	. . . 4 |- -. E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
8	7	nex 1546	. . 3 |- -. E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
9	8	abf 3535	. 2 |- { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) ) } = (/)
10	1, 2, 9	3eqtri 2254	1 |- ( x e. (/) , y e. B |-> C ) = (/)

Syntax hints:   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   (/)c0 3491   <.cop 3669   {coprab 6002   e. cmpo 6003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-oprab 6005  df-mpo 6006

This theorem is referenced by: (None)