Theorem nfmpo1 5878

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 5819	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
2		nfoprab1 5860	. 2 |- F/_ x { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
3	1, 2	nfcxfr 2293	1 |- F/_ x ( x e. A , y e. B |-> C )

Syntax hints:   /\ wa 103   = wceq 1332   e. wcel 2125   F/_wnfc 2283   {coprab 5815   e. cmpo 5816

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-oprab 5818  df-mpo 5819

This theorem is referenced by:  ovmpos  5934  ov2gf  5935  ovmpodxf  5936  ovmpodf  5942  ovmpodv2  5944  xpcomco  6760  mapxpen  6782  cnmpt21  12630  cnmpt2t  12632  cnmptcom  12637