Theorem nfmpo1 5985

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 5923	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
2		nfoprab1 5967	. 2 |- F/_ x { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
3	1, 2	nfcxfr 2333	1 |- F/_ x ( x e. A , y e. B |-> C )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   F/_wnfc 2323   {coprab 5919   e. cmpo 5920

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-oprab 5922  df-mpo 5923

This theorem is referenced by:  ovmpos  6042  ov2gf  6043  ovmpodxf  6044  ovmpodf  6050  ovmpodv2  6052  xpcomco  6880  mapxpen  6904  cnmpt21  14459  cnmpt2t  14461  cnmptcom  14466