Theorem mpo0 5947

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 5882	. 2 |- ( x e. (/) , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. (/) /\ y e. B ) /\ z = C ) }
2		df-oprab 5881	. 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 3428	. . . . . . 7 |- -. x e. (/)
4		simprll 537	. . . . . . 7 |- ( ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) ) -> x e. (/) )
5	3, 4	mto 662	. . . . . 6 |- -. ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
6	5	nex 1500	. . . . 5 |- -. E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
7	6	nex 1500	. . . 4 |- -. E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
8	7	nex 1500	. . 3 |- -. E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
9	8	abf 3468	. 2 |- { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) ) } = (/)
10	1, 2, 9	3eqtri 2202	1 |- ( x e. (/) , y e. B |-> C ) = (/)

Syntax hints:   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   (/)c0 3424   <.cop 3597   {coprab 5878   e. cmpo 5879

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425  df-oprab 5881  df-mpo 5882

This theorem is referenced by: (None)