Theorem nfmpo1 5909

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 5847	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
2		nfoprab1 5891	. 2 |- F/_ x { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
3	1, 2	nfcxfr 2305	1 |- F/_ x ( x e. A , y e. B |-> C )

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   F/_wnfc 2295   {coprab 5843   e. cmpo 5844

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-oprab 5846  df-mpo 5847

This theorem is referenced by:  ovmpos  5965  ov2gf  5966  ovmpodxf  5967  ovmpodf  5973  ovmpodv2  5975  xpcomco  6792  mapxpen  6814  cnmpt21  12931  cnmpt2t  12933  cnmptcom  12938