Theorem nfmpo1 5920

Description: Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)

Assertion

Ref	Expression
nfmpo1	|- F/_ x ( x e. A , y e. B |-> C )

Proof of Theorem nfmpo1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpo 5858	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
2		nfoprab1 5902	. 2 |- F/_ x { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
3	1, 2	nfcxfr 2309	1 |- F/_ x ( x e. A , y e. B |-> C )

Syntax hints:   /\ wa 103   = wceq 1348   e. wcel 2141   F/_wnfc 2299   {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-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-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-oprab 5857  df-mpo 5858

This theorem is referenced by:  ovmpos  5976  ov2gf  5977  ovmpodxf  5978  ovmpodf  5984  ovmpodv2  5986  xpcomco  6804  mapxpen  6826  cnmpt21  13085  cnmpt2t  13087  cnmptcom  13092