Theorem mpo0 6148

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 6080	. 2 |- ( x e. (/) , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. (/) /\ y e. B ) /\ z = C ) }
2		df-oprab 6079	. 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 3525	. . . . . . 7 |- -. x e. (/)
4		simprll 543	. . . . . . 7 |- ( ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) ) -> x e. (/) )
5	3, 4	mto 672	. . . . . 6 |- -. ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
6	5	nex 1553	. . . . 5 |- -. E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
7	6	nex 1553	. . . 4 |- -. E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
8	7	nex 1553	. . 3 |- -. E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) )
9	8	abf 3569	. 2 |- { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ( ( x e. (/) /\ y e. B ) /\ z = C ) ) } = (/)
10	1, 2, 9	3eqtri 2263	1 |- ( x e. (/) , y e. B |-> C ) = (/)

Syntax hints:   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   (/)c0 3520   <.cop 3708   {coprab 6076   e. cmpo 6077

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-oprab 6079  df-mpo 6080

This theorem is referenced by: (None)