Theorem mpo0 5923

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 5858	. 2 |- ( x e. (/) , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. (/) /\ y e. B ) /\ z = C ) }
2		df-oprab 5857	. 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 3418	. . . . . . 7 |- -. x e. (/)
4		simprll 532	. . . . . . 7 |- ( ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) ) -> x e. (/) )
5	3, 4	mto 657	. . . . . 6 |- -. ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
6	5	nex 1493	. . . . 5 |- -. E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
7	6	nex 1493	. . . 4 |- -. E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
8	7	nex 1493	. . 3 |- -. E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
9	8	abf 3458	. 2 |- { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) ) } = (/)
10	1, 2, 9	3eqtri 2195	1 |- ( x e. (/) , y e. B |-> C ) = (/)

Syntax hints:   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   (/)c0 3414   <.cop 3586   {coprab 5854   e. cmpo 5855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-oprab 5857  df-mpo 5858

This theorem is referenced by: (None)